diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Lattice.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Lattice.sym.json" new file mode 100644--- /dev/null +++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Lattice.sym.json" @@ -0,0 +1 @@ +[{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","smul"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3 : CommRing.{u_1} R] (A : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.22 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18))))] (M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) [inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.34 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.26 M] (a : Units.{u_2} A (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7))))), Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.26 (HSMul.hSMul.{u_2, u_3, u_3} (Units.{u_2} A (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7))))) (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (instHSMul.{u_2, u_3} (Units.{u_2} A (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7))))) (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (Units.instSMul.{u_2, u_3} A (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)))) (SMulZeroClass.toSMul.{u_2, u_3} A (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (AddZero.toZero.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (Submodule.pointwiseAddCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (Submodule.pointwiseAddCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18))) (DistribMulAction.toDistribSMul.{u_2, u_3} A (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18) (Submodule.pointwiseAddCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18)) (Submodule.pointwiseDistribMulAction.{u_2, u_1, u_3} A R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18 (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)))) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.22) (IsScalarTower.to_smulCommClass'.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.3) A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.4232235684._hygCtx._hyg.26))))))) a M)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n (M : Submodule R V) [Submodule.IsLattice A M] (a : Aˣ), Submodule.IsLattice A (a • M)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n (M : Submodule R V) [Submodule.IsLattice A M] (a : Aˣ), Submodule.IsLattice A (a • M)","typeReferences":[["IsScalarTower"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["IsScalarTower","to_smulCommClass'"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Units","instSMul"],["Units"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["CommRing"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Submodule","IsLattice"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["smulCommClass_self"],["Submodule","ext"],["Finset"],["Eq","trans"],["Submodule","smul_span"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule","span"],["Set","toFinite"],["Finite","of_fintype"],["Set","Elem"],["Submodule","coe_pointwise_smul"],["IsScalarTower","to_smulCommClass'"],["Eq","symm"],["DistribMulAction","toMulAction"],["Submodule","IsLattice","mk"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["Units"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["True","intro"],["AddZeroClass","toAddZero"],["FinsetCoe","fintype"],["Submodule","FG"],["Ring","toSemiring"],["Exists","casesOn"],["SetLike","coe"],["Units","val"],["HSMul","hSMul"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["id"],["Submodule","IsLattice"],["Eq","mpr"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["Submodule","fg_span"],["AddMonoid","toAddZeroClass"],["Finset","instSetLike"],["Units","smulCommClass_left"],["Units","instGroup"],["CommMonoid","toMonoid"],["Submodule","IsLattice","fg"],["Units","instInv"],["AddCommMonoid","toAddMonoid"],["Iff","intro"],["Submodule"],["congrArg"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Eq"],["Units","instSMul"],["Inv","inv"],["CommRing","toCommSemiring"],["True"],["Set"],["Submodule","IsLattice","span_eq_top"],["Submodule","instTop"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommRing","toCommMonoid"],["Finite","Set","finite_image"],["eq_self"],["CommRing","toRing"],["Units","instMonoid"],["Module","toDistribMulAction"],["of_eq_true"],["Units","instDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Units","instMulAction"],["smul_inv_smul"],["trivial"],["Submodule","smul_mem_pointwise_smul"]]},{"isProp":false,"kind":"definition","name":["Module","Basis","extendOfIsLattice"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.31 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10] {κ : Type.{u_4}} {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.39 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 M], (Module.Basis.{u_4, u_1, u_3} κ R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (Submodule.module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M)) -> (Module.Basis.{u_4, u_2, u_3} κ K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18)","typeFull":"{R : Type u_1} →\n [inst : CommRing R] →\n (K : Type u_2) →\n [inst_1 : Field K] →\n [inst_2 : Algebra R K] →\n {V : Type u_3} →\n [inst_3 : AddCommGroup V] →\n [inst_4 : Module K V] →\n [inst_5 : Module R V] →\n [inst_6 : IsScalarTower R K V] →\n [IsFractionRing R K] →\n {κ : Type u_4} →\n {M : Submodule R V} → [Submodule.IsLattice K M] → Module.Basis κ R ↥M → Module.Basis κ K V","typeReadable":"{R : Type u_1} →\n [inst : CommRing R] →\n (K : Type u_2) →\n [inst_1 : Field K] →\n [inst_2 : Algebra R K] →\n {V : Type u_3} →\n [inst_3 : AddCommGroup V] →\n [inst_4 : Module K V] →\n [inst_5 : Module R V] →\n [inst_6 : IsScalarTower R K V] →\n [IsFractionRing R K] →\n {κ : Type u_4} →\n {M : Submodule R V} → [Submodule.IsLattice K M] → Module.Basis κ R ↥M → Module.Basis κ K V","typeReferences":[["Semifield","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["IsScalarTower"],["Module"],["Field"],["Submodule","module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule"],["Field","toEuclideanDomain"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Module","Basis"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["CommRing"],["AddZeroClass","toAddZero"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["Submodule","IsLattice"],["IsFractionRing"],["Submodule","addCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Module","Basis","instFunLike"],["Module","Basis","extendOfIsLattice","_proof_5"],["CommRing","toCommSemiring"],["Subtype"],["SetLike","instMembership"],["Submodule","module"],["Membership","mem"],["CommSemiring","toSemiring"],["Module","Basis","extendOfIsLattice","_proof_4"],["Module","Basis","mk"],["DivisionSemiring","toSemiring"],["DFunLike","coe"],["Subtype","val"],["Submodule"],["Submodule","setLike"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["Module","Basis"],["Semifield","toDivisionSemiring"],["Submodule","addCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","rank'"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2874832418._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18))))] (M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) (N : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) [inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.40 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 M] [inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.44 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 N], Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.26 (Max.max.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) (SemilatticeSup.toMax.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) (Lattice.toSemilatticeSup.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) (ConditionallyCompleteLattice.toLattice.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18) (Submodule.completeLattice.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2659813699._hygCtx._hyg.18))))) M N)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n (M N : Submodule R V) [Submodule.IsLattice A M] [Submodule.IsLattice A N], Submodule.IsLattice A (M ⊔ N)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n (M N : Submodule R V) [Submodule.IsLattice A M] [Submodule.IsLattice A N], Submodule.IsLattice A (M ⊔ N)","typeReferences":[["Lattice","toSemilatticeSup"],["IsScalarTower"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["CompleteLattice","toConditionallyCompleteLattice"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["CommRing"],["Algebra"],["Module","toDistribMulAction"],["Max","max"],["SubNegMonoid","toAddMonoid"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["AddCommGroup","toAddCommMonoid"],["Submodule","IsLattice"],["ConditionallyCompleteLattice","toLattice"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Lattice","toSemilatticeSup"],["CommRing","toCommSemiring"],["Submodule","IsLattice","fg"],["Submodule","IsLattice","of_le_of_isLattice_of_fg"],["CommSemiring","toSemiring"],["le_sup_left"],["Submodule"],["Max","max"],["SemilatticeSup","toMax"],["AddCommGroup","toAddCommMonoid"],["Submodule","completeLattice"],["Submodule","FG","sup"],["ConditionallyCompleteLattice","toLattice"],["CompleteLattice","toConditionallyCompleteLattice"]]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","finrank_of_pi"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))] [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.31 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.34 : IsPrincipalIdealRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3))] {ι : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.38 : Fintype.{u_4} ι] [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.41 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10] (M : Submodule.{u_1, max u_2 u_4} R (ι -> K) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10))) [inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.52 : Submodule.IsLattice.{u_1, u_2, max u_2 u_4} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10 (ι -> K) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10)) (Pi.Function.module.{u_4, u_2, u_2} ι K K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Semiring.toModule.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))) (Pi.isScalarTower.{u_4, u_1, u_2, u_2} ι R K (fun (a._@._internal._hyg.0 : ι) => K) (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10) (fun (i : ι) => SemigroupAction.toSMul.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_2} K (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (MulAction.toSemigroupAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Semiring.toModule.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))) i)) i)) i))) (fun (i : ι) => SemigroupAction.toSMul.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3))))) (MulAction.toSemigroupAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10) i)) i)) i))) (fun (i : ι) => IsScalarTower.right.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10 (Pi.isScalarTower.{u_4, u_1, u_2, u_2} ι R K (fun (a._@._internal._hyg.0 : ι) => K) (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10) (fun (i : ι) => SemigroupAction.toSMul.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_2} K (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (MulAction.toSemigroupAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Semiring.toModule.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))) i)) i)) i))) (fun (i : ι) => SemigroupAction.toSMul.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3))))) (MulAction.toSemigroupAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10) i)) i)) i))) (fun (i : ι) => IsScalarTower.right.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10)) M], Eq.{1} Nat (Module.finrank.{u_1, max u_2 u_4} R (Subtype.{succ (max u_2 u_4)} (ι -> K) (fun (x : ι -> K) => Membership.mem.{max u_2 u_4, max u_2 u_4} (ι -> K) (Submodule.{u_1, max u_2 u_4} R (ι -> K) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K 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inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10))) (ι -> K) (Submodule.setLike.{u_1, max u_2 u_4} R (ι -> K) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10)))) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, max u_2 u_4} R (ι -> K) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10)) M) (Submodule.module.{u_1, max u_2 u_4} R (ι -> K) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.10)) M)) (Fintype.card.{u_4} ι inst._@.Mathlib.Algebra.Module.Lattice.1406385596._hygCtx._hyg.38)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] [IsDomain R]\n [IsPrincipalIdealRing R] {ι : Type u_4} [inst_5 : Fintype ι] [IsFractionRing R K] (M : Submodule R (ι → K))\n [Submodule.IsLattice K M], Module.finrank R ↥M = Fintype.card ι","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] [IsDomain R]\n [IsPrincipalIdealRing R] {ι : Type u_4} [inst_5 : Fintype ι] [IsFractionRing R K] (M : Submodule R (ι → K))\n [Submodule.IsLattice K M], Module.finrank R ↥M = Fintype.card ι","typeReferences":[["Semifield","toCommSemiring"],["Subtype"],["EuclideanDomain","toCommRing"],["Field"],["Submodule","module"],["Membership","mem"],["Pi","addCommMonoid"],["SemigroupAction","toSMul"],["IsDomain"],["AddCommMonoid","toAddMonoid"],["CommRing","toNonUnitalCommRing"],["Submodule"],["Pi","Function","module"],["Fintype","card"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Module","finrank"],["IsPrincipalIdealRing"],["Field","toEuclideanDomain"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Monoid","toSemigroup"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toModule"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["Fintype"],["CommRing"],["Pi","isScalarTower"],["Algebra"],["Nat"],["Module","toDistribMulAction"],["Submodule","setLike"],["Field","toSemifield"],["IsFractionRing"],["Submodule","IsLattice"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"]],"valueReferences":[["EuclideanDomain","toCommRing"],["Subtype"],["Submodule","module"],["Pi","addCommMonoid"],["Membership","mem"],["CommRing","toNonUnitalCommRing"],["Submodule"],["Pi","Function","module"],["Fintype","card"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Field","toEuclideanDomain"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["DivisionSemiring","toSemiring"],["Submodule","setLike"],["Submodule","IsLattice","rank_of_pi"],["Module","finrank_eq_of_rank_eq"],["Field","toSemifield"],["Submodule","addCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","inf"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.31 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.34 : IsPrincipalIdealRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.37 : Module.Finite.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.41 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10] (M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22) (N : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22) [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.51 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 M] [inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.55 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 N], Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.26 (Min.min.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22) (Submodule.instMin.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2823200984._hygCtx._hyg.22) M N)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] [IsDomain R]\n [IsPrincipalIdealRing R] [Module.Finite K V] [IsFractionRing R K] (M N : Submodule R V) [Submodule.IsLattice K M]\n [Submodule.IsLattice K N], Submodule.IsLattice K (M ⊓ N)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] [IsDomain R]\n [IsPrincipalIdealRing R] [Module.Finite K V] [IsFractionRing R K] (M N : Submodule R V) [Submodule.IsLattice K M]\n [Submodule.IsLattice K N], Submodule.IsLattice K (M ⊓ N)","typeReferences":[["Semifield","toCommSemiring"],["EuclideanDomain","toCommRing"],["IsScalarTower"],["Module"],["Field"],["AddCommGroup","toAddGroup"],["IsDomain"],["SMulZeroClass","toSMul"],["Submodule"],["IsPrincipalIdealRing"],["Field","toEuclideanDomain"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["CommRing"],["AddZeroClass","toAddZero"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Min","min"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["Submodule","IsLattice"],["IsFractionRing"],["Submodule","instMin"],["AddZero","toZero"],["Module","Finite"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["le_refl"],["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Membership","mem"],["Cardinal","eq_of_add_eq_add_left"],["Submodule","span"],["Cardinal","partialOrder"],["Submodule","subtype"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["EuclideanDomain","toNontrivial"],["Eq","symm"],["Module","IsNoetherian","finite"],["Submodule","IsLattice","mk"],["IsNoetherianRing","strongRankCondition"],["Semifield","toDivisionSemiring"],["Cardinal","instAdd"],["Module","Finite","iff_fg"],["SetLike","instMembership"],["Submodule","IsLattice","rank'"],["Submodule","addCommGroup"],["DivisionSemiring","toSemiring"],["Module","rank_lt_aleph0"],["Submodule","FG"],["Ring","toSemiring"],["Submodule","range_subtype"],["SetLike","coe"],["Module","rank"],["IsDomain","hasRankNullity"],["inf_le_left"],["AddCommGroup","toAddCommMonoid"],["id"],["Top","top"],["Eq","mpr"],["ConditionallyCompleteLattice","toLattice"],["Submodule","instMin"],["Submodule","IsLattice","finite"],["EuclideanDomain","toCommRing"],["Subtype"],["Eq","mp"],["Submodule","module"],["LinearMap","range"],["Submodule"],["congrArg"],["Field","toEuclideanDomain"],["isNoetherian_of_isNoetherianRing_of_finite"],["PrincipalIdealRing","isNoetherianRing"],["Eq"],["Submodule","IsLattice","sup"],["Cardinal","instLE"],["CompleteLattice","toConditionallyCompleteLattice"],["propext"],["CommRing","toCommSemiring"],["Lattice","toSemilatticeInf"],["instHAdd"],["DivisionRing","toRing"],["CommSemiring","toSemiring"],["Submodule","instTop"],["Field","toDivisionRing"],["Submodule","rank_sup_add_rank_inf_eq"],["Cardinal"],["RingHomSurjective","ids"],["HAdd","hAdd"],["CommRing","toRing"],["Max","max"],["EuclideanDomain","to_principal_ideal_domain"],["isNoetherian_of_le"],["Submodule","setLike"],["Min","min"],["SemilatticeSup","toMax"],["LE","le"],["Field","toSemifield"],["Submodule","completeLattice"],["inferInstance"],["Submodule","span_range_eq_top_of_injective_of_rank_le"],["Submodule","injective_subtype"],["Submodule","addCommMonoid"],["Module","Finite"]]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","fg"],"typeFallback":"forall {R : Type.{u_1}} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R} {A : outParam.{succ (succ u_2)} Type.{u_2}} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))} {V : Type.{u_3}} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))} {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23} [self : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M], Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 M","typeFull":"∀ {R : Type u_1} {inst : CommRing R} {A : outParam (Type u_2)} {inst_1 : CommRing A} {inst_2 : Algebra R A}\n {V : Type u_3} {inst_3 : AddCommMonoid V} {inst_4 : Module R V} {inst_5 : Module A V} {inst_6 : IsScalarTower R A V}\n {inst_7 : Algebra R A} {inst_8 : IsScalarTower R A V} {M : Submodule R V} [self : Submodule.IsLattice A M], M.FG","typeReadable":"∀ {R : Type u_1} {inst : CommRing R} {A : outParam (Type u_2)} {inst_1 : CommRing A} {inst_2 : Algebra R A}\n {V : Type u_3} {inst_3 : AddCommMonoid V} {inst_4 : Module R V} {inst_5 : Module A V} {inst_6 : IsScalarTower R A V}\n {inst_7 : Algebra R A} {inst_8 : IsScalarTower R A V} {M : Submodule R V} [self : Submodule.IsLattice A M], M.FG","typeReferences":[["CommRing","toCommSemiring"],["Module"],["IsScalarTower"],["outParam"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["CommRing"],["AddZeroClass","toAddZero"],["Submodule","FG"],["Submodule"],["Algebra"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["Submodule","IsLattice"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","span_eq_top"],"typeFallback":"forall {R : Type.{u_1}} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R} {A : outParam.{succ (succ u_2)} Type.{u_2}} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))} {V : Type.{u_3}} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A 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(Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))} {inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, 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CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.31 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.34 : IsPrincipalIdealRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.37 : Module.IsTorsionFree.{u_1, u_2} R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.10)] (M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22) [inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.44 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.26 M], Module.Free.{u_1, u_3} R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22 M) (Submodule.module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2965627298._hygCtx._hyg.22 M)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] [IsDomain R]\n [IsPrincipalIdealRing R] [Module.IsTorsionFree R K] (M : Submodule R V) [Submodule.IsLattice K M], Module.Free R ↥M","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] [IsDomain R]\n [IsPrincipalIdealRing R] [Module.IsTorsionFree R K] (M : Submodule R V) [Submodule.IsLattice K M], Module.Free R ↥M","typeReferences":[["Module","IsTorsionFree"],["Subtype"],["EuclideanDomain","toCommRing"],["IsScalarTower"],["Module"],["Field"],["Submodule","module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["IsDomain"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["Submodule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsPrincipalIdealRing"],["Field","toEuclideanDomain"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["CommRing"],["AddZeroClass","toAddZero"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Module","Free"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["Submodule","IsLattice"],["Submodule","addCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Membership","mem"],["Ideal","instIsTorsionFreeSubtypeMemSubmodule"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["EuclideanDomain","toNontrivial"],["Semifield","toDivisionSemiring"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["GroupWithZero","toNoZeroSMulDivisors"],["Submodule","module","_proof_1"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","addCommGroup"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["Module","IsTorsionFree","to_faithfulSMul"],["AddZeroClass","toAddZero"],["AddCommGroup","toAddCommMonoid"],["IsDomain","toIsCancelMulZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Submodule","IsLattice","finite"],["Subtype"],["EuclideanDomain","toCommRing"],["Submodule","module"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toSubNegMonoid"],["Submodule"],["Field","toEuclideanDomain"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["CommRing","toCommSemiring"],["IsDomain","toNontrivial"],["instIsDomain"],["DivisionRing","toRing"],["CommSemiring","toSemiring"],["Field","toDivisionRing"],["Module","free_of_finite_type_torsion_free'"],["CommRing","toRing"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Field","toSemifield"],["Module","Free"],["inferInstance"],["Module","IsTorsionFree","trans_faithfulSMul"],["Submodule","addCommMonoid"],["instIsTorsionFreeOfIsDomainOfNoZeroSMulDivisors"]]},{"isProp":true,"kind":"theorem","name":["Module","Basis","extendOfIsLattice","eq_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.31 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10] {κ : Type.{u_4}} {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.39 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 M] (b : Module.Basis.{u_4, u_1, u_3} κ R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (Submodule.module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M)), Eq.{max (max (succ u_4) (succ u_2)) (succ u_3)} (Module.Basis.{u_4, u_2, u_3} κ K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18) (Module.Basis.extendOfIsLattice.{u_1, u_2, u_3, u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15 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[inst_5 : Module R V] [inst_6 : IsScalarTower R K V]\n [inst_7 : IsFractionRing R K] {κ : Type u_4} {M : Submodule R V} [inst_8 : Submodule.IsLattice K M]\n (b : Module.Basis κ R ↥M), Module.Basis.extendOfIsLattice K b = Module.Basis.mk ⋯ 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.31 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.10] {κ : Type.{u_4}} {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22} [inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.39 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.26 M] (b : Module.Basis.{u_4, u_1, u_3} κ R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) 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inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22 M) (Submodule.module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1041894146._hygCtx._hyg.22 M)) b k))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V]\n [inst_7 : IsFractionRing R K] {κ : Type u_4} {M : Submodule R V} [inst_8 : Submodule.IsLattice K M]\n (b : Module.Basis κ R ↥M) (k : κ), (Module.Basis.extendOfIsLattice K b) k = ↑(b k)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V]\n [inst_7 : IsFractionRing R K] {κ : Type u_4} {M : Submodule R V} [inst_8 : Submodule.IsLattice K M]\n (b : Module.Basis κ R ↥M) (k : κ), (Module.Basis.extendOfIsLattice K b) k = ↑(b 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{R : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 : CommRing.{u_4} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 : Algebra.{u_4, u_2} R K (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)))] {V : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15 : AddCommGroup.{u_1} V] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 : Module.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 : Module.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 : IsScalarTower.{u_4, u_2, u_1} R K V (Algebra.toSMul.{u_4, u_2} R K (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_1} K V (AddZero.toZero.{u_1} V (AddZeroClass.toAddZero.{u_1} V (AddMonoid.toAddZeroClass.{u_1} V (SubNegMonoid.toAddMonoid.{u_1} V (AddGroup.toSubNegMonoid.{u_1} V (AddCommGroup.toAddGroup.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K V (AddMonoid.toAddZeroClass.{u_1} V (SubNegMonoid.toAddMonoid.{u_1} V (AddGroup.toSubNegMonoid.{u_1} V (AddCommGroup.toAddGroup.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_1} V (AddGroup.toSubNegMonoid.{u_1} V (AddCommGroup.toAddGroup.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_4, u_1} R V (AddZero.toZero.{u_1} V (AddZeroClass.toAddZero.{u_1} V (AddMonoid.toAddZeroClass.{u_1} V (SubNegMonoid.toAddMonoid.{u_1} V (AddGroup.toSubNegMonoid.{u_1} V (AddCommGroup.toAddGroup.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_4, u_1} R V (AddMonoid.toAddZeroClass.{u_1} V (SubNegMonoid.toAddMonoid.{u_1} V (AddGroup.toSubNegMonoid.{u_1} V (AddCommGroup.toAddGroup.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_4, u_1} R V (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_1} V (AddGroup.toSubNegMonoid.{u_1} V (AddCommGroup.toAddGroup.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22))))] {κ : Type.{u_3}} {M : Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.39 : Submodule.IsLattice.{u_4, u_2, u_1} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 M] (b : Module.Basis.{u_3, u_4, u_1} κ R (Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (Submodule.module.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M)), LE.le.{u_1} (Submodule.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18) (Submodule.instPartialOrder.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18))) (Top.top.{u_1} (Submodule.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18) (Submodule.instTop.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18)) (Submodule.span.{u_2, u_1} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 (Set.range.{u_1, succ u_3} V κ (fun (i : κ) => Function.comp.{succ u_3, succ u_1, succ u_1} κ (Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) V (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{u_4, u_4, u_1, u_1} R R (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (RingHom.id.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)))) (Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) V (Submodule.addCommMonoid.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) (Submodule.module.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) => V) (LinearMap.instFunLike.{u_4, u_4, u_1, u_1} R R (Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) (Submodule.module.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 (RingHom.id.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3))))) (Submodule.subtype.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M)) (DFunLike.coe.{max (max (succ u_4) (succ u_1)) (succ u_3), succ u_3, succ u_1} (Module.Basis.{u_3, u_4, u_1} κ R (Subtype.{succ u_1} V (fun (x : V) => Membership.mem.{u_1, u_1} V (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_1} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_1} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) 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Algebra R K] {V : Type u_1}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] {κ : Type u_3}\n {M : Submodule R V} [Submodule.IsLattice K M] (b : Module.Basis κ R ↥M),\n ⊤ ≤ Submodule.span K (Set.range fun i => (⇑M.subtype ∘ ⇑b) i)","typeReadable":"∀ {R : Type u_4} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_1}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] {κ : Type u_3}\n {M : Submodule R V} [Submodule.IsLattice K M] (b : Module.Basis κ R ↥M),\n ⊤ ≤ Submodule.span K (Set.range fun i => (⇑M.subtype ∘ ⇑b) 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23} {motive : (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M) -> Sort.{u}} (t : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M), (forall (fg : Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 M) (span_eq_top : Eq.{succ u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.span.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) M)) (Top.top.{u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.instTop.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27))), motive (Submodule.IsLattice.mk.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M fg span_eq_top)) -> (motive t)","typeFull":"{R : Type u_1} →\n [inst : CommRing R] →\n {A : Type u_2} →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [inst_6 : IsScalarTower R A V] →\n [inst_7 : Algebra R A] →\n [inst_8 : IsScalarTower R A V] →\n {M : Submodule R V} →\n {motive : Submodule.IsLattice A M → Sort u} →\n (t : Submodule.IsLattice A M) →\n ((fg : M.FG) → (span_eq_top : Submodule.span A ↑M = ⊤) → motive ⋯) → motive t","typeReadable":"{R : Type u_1} →\n [inst : CommRing R] →\n {A : Type u_2} →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [inst_6 : IsScalarTower R A V] →\n [inst_7 : Algebra R A] →\n [inst_8 : IsScalarTower R A V] →\n {M : Submodule R V} →\n {motive : Submodule.IsLattice A M → Sort u} →\n (t : Submodule.IsLattice A M) →\n ((fg : M.FG) → (span_eq_top : Submodule.span A ↑M = ⊤) → motive ⋯) → motive t","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Submodule","IsLattice","mk"],["Eq"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["Submodule","instTop"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["Submodule","FG"],["Algebra"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Top","top"],["Submodule","IsLattice"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","IsLattice","rec"]]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","rank_of_pi"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))] [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.31 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.34 : IsPrincipalIdealRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3))] {ι : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.38 : Fintype.{u_4} ι] [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.41 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10] (M : Submodule.{u_1, max u_2 u_4} R (ι -> K) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10))) [inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.52 : Submodule.IsLattice.{u_1, u_2, max u_2 u_4} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10 (ι -> K) (Pi.addCommMonoid.{u_4, u_2} ι (fun (a._@._internal._hyg.0 : ι) => K) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))))) (Pi.Function.module.{u_4, u_1, u_2} ι R K (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) (Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10)) (Pi.Function.module.{u_4, u_2, u_2} ι K K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) (Semiring.toModule.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))) (Pi.isScalarTower.{u_4, u_1, u_2, u_2} ι R K (fun (a._@._internal._hyg.0 : ι) => K) (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10) (fun (i : ι) => SemigroupAction.toSMul.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_2} K (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) (MulAction.toSemigroupAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Semiring.toModule.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))) i)) i)) i))) (fun (i : ι) => SemigroupAction.toSMul.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3))))) (MulAction.toSemigroupAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10) i)) i)) i))) (fun (i : ι) => IsScalarTower.right.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10)) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10 (Pi.isScalarTower.{u_4, u_1, u_2, u_2} ι R K (fun (a._@._internal._hyg.0 : ι) => K) (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10) (fun (i : ι) => SemigroupAction.toSMul.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_2} K (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) (MulAction.toSemigroupAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_2, u_2} K ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7)))) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Semiring.toModule.{u_2} K (CommSemiring.toSemiring.{u_2} K (CommRing.toCommSemiring.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))) i)) i)) i))) (fun (i : ι) => SemigroupAction.toSMul.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (Monoid.toSemigroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3))))) (MulAction.toSemigroupAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)))) ((fun (i : ι) => DistribMulAction.toMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)))) ((fun (i : ι) => AddCommMonoid.toAddMonoid.{u_2} ((fun (a._@._internal._hyg.0 : ι) => K) i) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i)) i) ((fun (i : ι) => Module.toDistribMulAction.{u_1, u_2} R ((fun (a._@._internal._hyg.0 : ι) => K) i) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3)) ((fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} K (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} K (CommRing.toNonUnitalCommRing.{u_2} K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))))))) i) ((fun (i : ι) => Algebra.toModule.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10) i)) i)) i))) (fun (i : ι) => IsScalarTower.right.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.301385580._hygCtx._hyg.10)) M], Eq.{max (succ (succ u_2)) (succ 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3 : CommRing.{u_1} R] (A : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.22 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18))))] (M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18) [inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.34 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.26 M], Module.Finite.{u_1, u_3} R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18)) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18 M) (Submodule.module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1088375812._hygCtx._hyg.18 M)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n (M : Submodule R V) [Submodule.IsLattice A M], Module.Finite R ↥M","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n (M : Submodule R V) [Submodule.IsLattice A M], Module.Finite R ↥M","typeReferences":[["Subtype"],["IsScalarTower"],["Module"],["Submodule","module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["CommRing"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Submodule","IsLattice"],["Submodule","addCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Module","Finite"]],"valueReferences":[["Module","Finite","iff_fg"],["CommRing","toCommSemiring"],["Subtype"],["SetLike","instMembership"],["Submodule","IsLattice","fg"],["Submodule","module"],["Membership","mem"],["CommSemiring","toSemiring"],["Submodule","FG"],["congrArg"],["Submodule"],["Submodule","setLike"],["id"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["Eq"],["Submodule","addCommMonoid"],["propext"],["Module","Finite"]]},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","of_le_of_isLattice_of_fg"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3 : CommRing.{u_1} R] (A : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.22 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18))))] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18} {N : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18}, (LE.le.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18) (Preorder.toLE.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18) (PartialOrder.toPreorder.{u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18) (Submodule.instPartialOrder.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18))) M N) -> (forall [inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.45 : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.26 M], (Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18 N) -> (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.3 A inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.1876672213._hygCtx._hyg.26 N))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n {M N : Submodule R V}, M ≤ N → ∀ [Submodule.IsLattice A M], N.FG → Submodule.IsLattice A N","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (A : Type u_2) [inst_1 : CommRing A] [inst_2 : Algebra R A] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n {M N : Submodule R V}, M ≤ N → ∀ [Submodule.IsLattice A M], N.FG → Submodule.IsLattice A 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R] (A : outParam.{succ (succ u_2)} Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))], (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) -> Prop","typeFull":"{R : Type u_1} →\n [inst : CommRing R] →\n (A : outParam (Type u_2)) →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [IsScalarTower R A V] → [inst_7 : Algebra R A] → [IsScalarTower R A V] → Submodule R V → Prop","typeReadable":"{R : Type u_1} →\n [inst : CommRing R] →\n (A : outParam (Type u_2)) →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [IsScalarTower R A V] → [inst_7 : Algebra R A] → [IsScalarTower R A V] → Submodule R V → 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[inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23} {motive : (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M) -> Sort.{u}}, (forall (fg : Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 M) (span_eq_top : Eq.{succ u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.span.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) M)) (Top.top.{u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.instTop.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27))), motive (Submodule.IsLattice.mk.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M fg span_eq_top)) -> (forall (t : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M), motive t)","typeFull":"{R : Type u_1} →\n [inst : CommRing R] →\n {A : Type u_2} →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [inst_6 : IsScalarTower R A V] →\n [inst_7 : Algebra R A] →\n [inst_8 : IsScalarTower R A V] →\n {M : Submodule R V} →\n {motive : Submodule.IsLattice A M → Sort u} →\n ((fg : M.FG) → (span_eq_top : Submodule.span A ↑M = ⊤) → motive ⋯) →\n (t : Submodule.IsLattice A M) → motive t","typeReadable":"{R : Type u_1} →\n [inst : CommRing R] →\n {A : Type u_2} →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [inst_6 : IsScalarTower R A V] →\n [inst_7 : Algebra R A] →\n [inst_8 : IsScalarTower R A V] →\n {M : Submodule R V} →\n {motive : Submodule.IsLattice A M → Sort u} →\n ((fg : M.FG) → (span_eq_top : Submodule.span A ↑M = ⊤) → motive ⋯) →\n (t : Submodule.IsLattice A M) → motive t","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Submodule","IsLattice","mk"],["Eq"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["Submodule","instTop"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["Submodule","FG"],["Algebra"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Top","top"],["Submodule","IsLattice"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":true,"kind":"constructor","name":["Submodule","IsLattice","mk"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R] {A : outParam.{succ (succ u_2)} Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23}, (Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 M) -> (Eq.{succ u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.span.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) M)) (Top.top.{u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.instTop.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27))) -> (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] {A : outParam (Type u_2)} [inst_1 : CommRing A] [inst_2 : Algebra R A]\n {V : Type u_3} [inst_3 : AddCommMonoid V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n [inst_7 : Algebra R A] [inst_8 : IsScalarTower R A V] {M : Submodule R V},\n M.FG → Submodule.span A ↑M = ⊤ → Submodule.IsLattice A M","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] {A : outParam (Type u_2)} [inst_1 : CommRing A] [inst_2 : Algebra R A]\n {V : Type u_3} [inst_3 : AddCommMonoid V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n [inst_7 : Algebra R A] [inst_8 : IsScalarTower R A V] {M : Submodule R V},\n M.FG → Submodule.span A ↑M = ⊤ → Submodule.IsLattice A M","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["outParam"],["Submodule","instTop"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["Submodule","FG"],["Algebra"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Top","top"],["Submodule","IsLattice"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Submodule","IsLattice","of_rank_le"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3 : CommRing.{u_1} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.26 : IsScalarTower.{u_1, u_2, u_3} R K V (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.31 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.34 : Module.Finite.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.38 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.10] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22}, (Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22 M) -> (LE.le.{succ u_3} Cardinal.{u_3} Cardinal.instLE.{u_3} (Module.rank.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.18) (Module.rank.{u_1, u_3} R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22 M) (Submodule.module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22 M))) -> (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.3 K (EuclideanDomain.toCommRing.{u_2} K (Field.toEuclideanDomain.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.10 V (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Lattice.241290039._hygCtx._hyg.26 M)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] [IsDomain R]\n [Module.Finite K V] [IsFractionRing R K] {M : Submodule R V},\n M.FG → Module.rank K V ≤ Module.rank R ↥M → Submodule.IsLattice K M","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [inst_6 : IsScalarTower R K V] [IsDomain R]\n [Module.Finite K V] [IsFractionRing R K] {M : Submodule R V},\n M.FG → Module.rank K V ≤ Module.rank R ↥M → Submodule.IsLattice K M","typeReferences":[["Semifield","toCommSemiring"],["EuclideanDomain","toCommRing"],["Subtype"],["IsScalarTower"],["Module"],["Field"],["Submodule","module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["IsDomain"],["SMulZeroClass","toSMul"],["Submodule"],["Field","toEuclideanDomain"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Cardinal","instLE"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["CommRing"],["AddZeroClass","toAddZero"],["Cardinal"],["Submodule","FG"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Module","rank"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["Submodule","IsLattice"],["IsFractionRing"],["Submodule","addCommMonoid"],["AddZero","toZero"],["Module","Finite"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["EuclideanDomain","toCommRing"],["Subtype"],["Eq","mp"],["LinearMap","range"],["Submodule","module"],["Membership","mem"],["Submodule","span"],["congrArg"],["Submodule","subtype"],["Submodule"],["Semiring","toNonAssocSemiring"],["Field","toEuclideanDomain"],["RingHom","id"],["Submodule","IsLattice","mk"],["congrFun'"],["Semifield","toDivisionSemiring"],["Eq"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["Set"],["Submodule","instTop"],["CommSemiring","toSemiring"],["Submodule","addCommGroup"],["DivisionSemiring","toSemiring"],["RingHomSurjective","ids"],["Submodule","range_subtype"],["CommRing","toRing"],["SetLike","coe"],["Submodule","setLike"],["Top","top"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["Submodule","span_range_eq_top_of_injective_of_rank_le"],["Submodule","injective_subtype"]]},{"isProp":true,"kind":"theorem","name":["Submodule","span_range_eq_top_of_injective_of_rank_le"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3 : CommRing.{u_1} R] {K : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.10 : Algebra.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7)))] {M : Type.{u}} {N : Type.{u}} [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.16 : IsDomain.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.19 : IsFractionRing.{u_1, u_2} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23 : AddCommGroup.{u} M] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.26 : Module.{u_1, u} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} M inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23)] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30 : AddCommGroup.{u} N] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33 : Module.{u_1, u} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30)] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37 : Module.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30)] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.41 : IsScalarTower.{u_1, u_2, u} R K N (Algebra.toSMul.{u_1, u_2} R K (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u} K N (AddZero.toZero.{u} N (AddZeroClass.toAddZero.{u} N (AddMonoid.toAddZeroClass.{u} N (SubNegMonoid.toAddMonoid.{u} N (AddGroup.toSubNegMonoid.{u} N (AddCommGroup.toAddGroup.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30)))))) (DistribSMul.toSMulZeroClass.{u_2, u} K N (AddMonoid.toAddZeroClass.{u} N (SubNegMonoid.toAddMonoid.{u} N (AddGroup.toSubNegMonoid.{u} N (AddCommGroup.toAddGroup.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30)))) (DistribMulAction.toDistribSMul.{u_2, u} K N (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u} N (AddGroup.toSubNegMonoid.{u} N (AddCommGroup.toAddGroup.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30))) (Module.toDistribMulAction.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37)))) (SMulZeroClass.toSMul.{u_1, u} R N (AddZero.toZero.{u} N (AddZeroClass.toAddZero.{u} N (AddMonoid.toAddZeroClass.{u} N (SubNegMonoid.toAddMonoid.{u} N (AddGroup.toSubNegMonoid.{u} N (AddCommGroup.toAddGroup.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30)))))) (DistribSMul.toSMulZeroClass.{u_1, u} R N (AddMonoid.toAddZeroClass.{u} N (SubNegMonoid.toAddMonoid.{u} N (AddGroup.toSubNegMonoid.{u} N (AddCommGroup.toAddGroup.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30)))) (DistribMulAction.toDistribSMul.{u_1, u} R N (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u} N (AddGroup.toSubNegMonoid.{u} N (AddCommGroup.toAddGroup.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30))) (Module.toDistribMulAction.{u_1, u} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33))))] [inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.46 : Module.Finite.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37] {f : LinearMap.{u_1, u_1, u, u} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)))) M N (AddCommGroup.toAddCommMonoid.{u} M inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33}, (Function.Injective.{succ u, succ u} M N (DFunLike.coe.{succ u, succ u, succ u} (LinearMap.{u_1, u_1, u, u} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)))) M N (AddCommGroup.toAddCommMonoid.{u} M inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (LinearMap.instFunLike.{u_1, u_1, u, u} R R M N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} M inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3))))) f)) -> (LE.le.{succ u} Cardinal.{u} Cardinal.instLE.{u} (Module.rank.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37) (Module.rank.{u_1, u} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} M inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.26)) -> (Eq.{succ u} (Submodule.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37) (Submodule.span.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37 (SetLike.coe.{u, u} (Submodule.{u_1, u} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33) N (Submodule.setLike.{u_1, u} R N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33) (LinearMap.range.{u_1, u_1, u, u} R R M N (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u} M inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.23) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.33 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3)))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.3))) f))) (Top.top.{u} (Submodule.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37) (Submodule.instTop.{u_2, u} K N (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u} N inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.30) inst._@.Mathlib.Algebra.Module.Lattice.931415000._hygCtx._hyg.37)))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : Field K] [inst_2 : Algebra R K] {M N : Type u}\n [IsDomain R] [IsFractionRing R K] [inst_5 : AddCommGroup M] [inst_6 : Module R M] [inst_7 : AddCommGroup N]\n [inst_8 : Module R N] [inst_9 : Module K N] [IsScalarTower R K N] [Module.Finite K N] {f : M →ₗ[R] N},\n Function.Injective ⇑f → Module.rank K N ≤ Module.rank R M → Submodule.span K ↑f.range = ⊤","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : Field K] [inst_2 : Algebra R K] {M N : Type u}\n [IsDomain R] [IsFractionRing R K] [inst_5 : AddCommGroup M] [inst_6 : Module R M] [inst_7 : AddCommGroup N]\n [inst_8 : Module R N] [inst_9 : Module K N] [IsScalarTower R K N] [Module.Finite K N] {f : M →ₗ[R] N},\n Function.Injective ⇑f → Module.rank K N ≤ Module.rank R M → Submodule.span K ↑f.range = ⊤","typeReferences":[["Semifield","toCommSemiring"],["LinearMap","instFunLike"],["Module"],["Field"],["IsScalarTower"],["LinearMap","range"],["AddCommGroup","toAddGroup"],["Submodule","span"],["IsDomain"],["SMulZeroClass","toSMul"],["DFunLike","coe"],["Submodule"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Eq"],["Cardinal","instLE"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["Submodule","instTop"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["LinearMap"],["CommRing"],["AddZeroClass","toAddZero"],["Cardinal"],["RingHomSurjective","ids"],["Algebra"],["Module","toDistribMulAction"],["SetLike","coe"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Module","rank"],["LE","le"],["Top","top"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["IsFractionRing"],["AddZero","toZero"],["Function","Injective"],["Module","Finite"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["Membership","mem"],["Submodule","span"],["Subtype","val"],["Set","Elem"],["Cardinal","mk"],["Cardinal","partialOrder"],["eq_top_iff"],["Module","finrank"],["Semiring","toNonAssocSemiring"],["LinearIndependent","cardinal_le_rank"],["RingHom","id"],["EuclideanDomain","toNontrivial"],["Eq","symm"],["IsNoetherianRing","strongRankCondition"],["Semifield","toDivisionSemiring"],["Eq","ndrec"],["Eq","rec"],["Submodule","span_mono"],["SetLike","instMembership"],["Exists"],["LinearMap","ker"],["And"],["DivisionSemiring","toSemiring"],["Fintype"],["LinearMap"],["Bot","bot"],["Set","instMembership"],["Ring","toSemiring"],["Exists","casesOn"],["Nat"],["SetLike","coe"],["LinearIndependent","iff_fractionRing"],["Module","rank"],["Iff","mpr"],["Submodule","instBot"],["id"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["IsDomain","hasRankNullity"],["LinearMap","ker_eq_bot"],["Eq","mpr"],["exists_apply_eq_apply","_simp_1"],["Nat","cast"],["EuclideanDomain","toCommRing"],["LinearMap","instFunLike"],["Eq","mp"],["LinearMap","range"],["exists_set_linearIndependent"],["DFunLike","coe"],["Set","range"],["LinearIndependent"],["congrArg"],["Submodule"],["Fintype","card"],["Field","toEuclideanDomain"],["DivisionRing","toDivisionSemiring"],["PrincipalIdealRing","isNoetherianRing"],["Preorder","toLE"],["Eq"],["le_antisymm"],["propext"],["Submodule","instOrderTop"],["CommRing","toCommSemiring"],["LinearMap","mem_range","_simp_1"],["True"],["Set"],["Submodule","instPartialOrder"],["DivisionRing","toRing"],["CommSemiring","toSemiring"],["Submodule","instTop"],["Field","toDivisionRing"],["Function","comp"],["Cardinal"],["Cardinal","instNatCast"],["RingHomSurjective","ids"],["Module","finrank_eq_rank"],["CommRing","toRing"],["LinearIndependent","span_eq_top_of_card_eq_finrank'"],["EuclideanDomain","to_principal_ideal_domain"],["SetLike","mem_coe","_simp_1"],["of_eq_true"],["Cardinal","mk_eq_nat_iff_fintype"],["Submodule","setLike"],["LE","le"],["Field","toSemifield"],["LinearIndependent","map'"],["OrderTop","toTop"],["And","casesOn"],["Function","Injective"]]},{"isProp":false,"kind":"definition","name":["Submodule","IsLattice","casesOn"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R] {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23} {motive : (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M) -> Sort.{u}} (t : Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M), (forall (fg : Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 M) (span_eq_top : Eq.{succ u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.span.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) M)) (Top.top.{u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.instTop.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27))), motive (Submodule.IsLattice.mk.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M fg span_eq_top)) -> (motive t)","typeFull":"{R : Type u_1} →\n [inst : CommRing R] →\n {A : Type u_2} →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [inst_6 : IsScalarTower R A V] →\n [inst_7 : Algebra R A] →\n [inst_8 : IsScalarTower R A V] →\n {M : Submodule R V} →\n {motive : Submodule.IsLattice A M → Sort u} →\n (t : Submodule.IsLattice A M) →\n ((fg : M.FG) → (span_eq_top : Submodule.span A ↑M = ⊤) → motive ⋯) → motive t","typeReadable":"{R : Type u_1} →\n [inst : CommRing R] →\n {A : Type u_2} →\n [inst_1 : CommRing A] →\n [inst_2 : Algebra R A] →\n {V : Type u_3} →\n [inst_3 : AddCommMonoid V] →\n [inst_4 : Module R V] →\n [inst_5 : Module A V] →\n [inst_6 : IsScalarTower R A V] →\n [inst_7 : Algebra R A] →\n [inst_8 : IsScalarTower R A V] →\n {M : Submodule R V} →\n {motive : Submodule.IsLattice A M → Sort u} →\n (t : Submodule.IsLattice A M) →\n ((fg : M.FG) → (span_eq_top : Submodule.span A ↑M = ⊤) → motive ⋯) → motive t","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Submodule","IsLattice","mk"],["Eq"],["DistribSMul","toSMulZeroClass"],["CommRing","toCommSemiring"],["Submodule","instTop"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["CommRing"],["Submodule","FG"],["Algebra"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Top","top"],["Submodule","IsLattice"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","IsLattice","rec"]]},{"isProp":true,"kind":"theorem","name":["Module","Basis","extendOfIsLattice","_proof_4"],"typeFallback":"forall {R : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3 : CommRing.{u_4} R] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7 : Field.{u_2} K] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10 : Algebra.{u_4, u_2} R K (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15 : AddCommGroup.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18 : Module.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 : Module.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.26 : IsScalarTower.{u_4, u_2, u_3} R K V (Algebra.toSMul.{u_4, u_2} R K (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10) (SMulZeroClass.toSMul.{u_2, u_3} K V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} K V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} K V (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} K V (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_4, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_4, u_3} R V (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)))) (SubNegMonoid.toAddMonoid.{u_3} V (AddGroup.toSubNegMonoid.{u_3} V (AddCommGroup.toAddGroup.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22))))] [inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.31 : IsFractionRing.{u_4, u_2} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3) K (Semifield.toCommSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.10] {κ : Type.{u_1}} {M : Submodule.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22} (b : Module.Basis.{u_1, u_4, u_3} κ R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x)) (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (Submodule.module.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M)), LinearIndependent.{u_1, u_2, u_3} κ K V (fun (i : κ) => Subtype.val.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) V (Submodule.setLike.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22)) M x) (DFunLike.coe.{max (max (succ u_4) (succ u_3)) (succ u_1), succ u_1, succ u_3} (Module.Basis.{u_1, u_4, u_3} κ R (Subtype.{succ u_3} V (fun (x : V) => Membership.mem.{u_3, u_3} V (Submodule.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22) 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(CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (Submodule.addCommMonoid.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M) (Submodule.module.{u_4, u_3} R V (CommSemiring.toSemiring.{u_4} R (CommRing.toCommSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.3)) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.22 M)) b i)) (DivisionSemiring.toSemiring.{u_2} K (Semifield.toDivisionSemiring.{u_2} K (Field.toSemifield.{u_2} K inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.7))) (AddCommGroup.toAddCommMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Lattice.2758782243._hygCtx._hyg.18","typeFull":"∀ {R : Type u_4} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [IsScalarTower R K V] [IsFractionRing R K]\n {κ : Type u_1} {M : Submodule R V} (b : Module.Basis κ R ↥M), LinearIndependent K fun i => ↑(b i)","typeReadable":"∀ {R : Type u_4} [inst : CommRing R] (K : Type u_2) [inst_1 : Field K] [inst_2 : Algebra R K] {V : Type u_3}\n [inst_3 : AddCommGroup V] [inst_4 : Module K V] [inst_5 : Module R V] [IsScalarTower R K V] [IsFractionRing R K]\n {κ : Type u_1} {M : Submodule R V} (b : Module.Basis κ R ↥M), LinearIndependent K fun i => ↑(b i)","typeReferences":[["Semifield","toCommSemiring"],["Subtype"],["IsScalarTower"],["Module"],["Field"],["Submodule","module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["DFunLike","coe"],["Subtype","val"],["LinearIndependent"],["Submodule"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Module","Basis"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["Module","Basis","instFunLike"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["CommSemiring","toSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["CommRing"],["AddZeroClass","toAddZero"],["Algebra"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["IsFractionRing"],["Submodule","addCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["_private","Mathlib","Algebra","Module","Lattice",0,"Module","Basis","extendOfIsLattice","_simp_3"],["SubtractionMonoid","toSubNegZeroMonoid"],["Module","Basis","linearIndependent"],["Ring","toNonAssocRing"],["Finset"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Iff","mp"],["SMulZeroClass","toSMul"],["Algebra","id"],["Subtype","val"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["DistribMulAction","toMulAction"],["Eq","symm"],["Finset","sum"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["Module","Basis","instFunLike"],["SetLike","instMembership"],["Submodule","addCommGroup"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["_private","Mathlib","Algebra","Module","Lattice",0,"Module","Basis","extendOfIsLattice","_simp_2"],["LinearIndependent","iff_fractionRing"],["Eq","refl"],["HSMul","hSMul"],["id"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Finset","instSetLike"],["EuclideanDomain","toCommRing"],["Subtype"],["Eq","mp"],["Submodule","module"],["linearIndependent_iff'"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["SubNegZeroMonoid","toNegZeroClass"],["LinearIndependent"],["congrArg"],["Submodule"],["Finset","sum_congr"],["Field","toEuclideanDomain"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["Module","Basis"],["Eq"],["propext"],["CommRing","toCommSemiring"],["IsScalarTower","left"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["CommRing","toRing"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Submodule","zero"],["Submodule","smul"],["Submodule","addCommMonoid"]]},{"isProp":true,"kind":"definition","name":["Submodule","IsLattice","mk","_flat_ctor"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 : CommRing.{u_1} R] {A : outParam.{succ (succ u_2)} Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] {V : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 : AddCommMonoid.{u_3} V] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 : Module.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 : Module.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 : IsScalarTower.{u_1, u_2, u_3} R A V (Algebra.toSMul.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36) (SMulZeroClass.toSMul.{u_2, u_3} A V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} A V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_2, u_3} A V (MonoidWithZero.toMonoid.{u_2} A (Semiring.toMonoidWithZero.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_3} R V (AddZero.toZero.{u_3} V (AddZeroClass.toAddZero.{u_3} V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R V (AddMonoid.toAddZeroClass.{u_3} V (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20)) (DistribMulAction.toDistribSMul.{u_1, u_3} R V (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)))) (AddCommMonoid.toAddMonoid.{u_3} V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20) (Module.toDistribMulAction.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23))))] {M : Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23}, (Submodule.FG.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 M) -> (Eq.{succ u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.span.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) V (Submodule.setLike.{u_1, u_3} R V (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23) M)) (Top.top.{u_3} (Submodule.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27) (Submodule.instTop.{u_2, u_3} A V (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27))) -> (Submodule.IsLattice.{u_1, u_2, u_3} R inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.7 A inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.15 V inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.Lattice.1206243495._hygCtx._hyg.40 M)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] {A : outParam (Type u_2)} [inst_1 : CommRing A] [inst_2 : Algebra R A]\n {V : Type u_3} [inst_3 : AddCommMonoid V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n [inst_7 : Algebra R A] [inst_8 : IsScalarTower R A V] {M : Submodule R V},\n M.FG → Submodule.span A ↑M = ⊤ → Submodule.IsLattice A M","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] {A : outParam (Type u_2)} [inst_1 : CommRing A] [inst_2 : Algebra R A]\n {V : Type u_3} [inst_3 : AddCommMonoid V] [inst_4 : Module R V] [inst_5 : Module A V] [inst_6 : IsScalarTower R A V]\n [inst_7 : Algebra R A] [inst_8 : IsScalarTower R A V] {M : Submodule R V},\n M.FG → Submodule.span A ↑M = ⊤ → Submodule.IsLattice A 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