diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Bilinear.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Bilinear.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Bilinear.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Submodule","apply_mem_map₂"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27))))) {m : M} {n : N} {p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19} {q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23}, (Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19)) p m) -> (Membership.mem.{u_3, u_3} N (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23) N (Submodule.setLike.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23)) q n) -> (Membership.mem.{u_4, u_4} P (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) (SetLike.instMembership.{u_4, u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) P (Submodule.setLike.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27)) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 f p q) (DFunLike.coe.{max (succ u_3) (succ u_4), succ u_3, succ u_4} (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) N (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : N) => P) (LinearMap.instFunLike.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7)))) (DFunLike.coe.{max (max (succ u_2) (succ u_3)) (succ u_4), succ u_2, max (succ u_3) (succ u_4)} (LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27))))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) (LinearMap.instFunLike.{u_1, u_1, u_2, max u_3 u_4} R R M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2787810736._hygCtx._hyg.7) 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{R : Type.{u_1}} {M : Type.{u_4}} [inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 : Semiring.{u_1} R] [inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15 : Module.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12] {ι : Sort.{u_9}} (s : ι -> (Set.{u_4} M)), Eq.{succ u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (iSup.{u_4, u_9} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (CompleteLattice.toConditionallyCompleteLattice.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (Submodule.completeLattice.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15))))) (fun (i : ι) => Submodule.span.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15 (s i))) (Submodule.span.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15 (Set.iUnion.{u_4, u_9} M ι (fun (i : ι) => s i)))","typeFull":"∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_9}\n  (s : ι → Set M), ⨆ i, Submodule.span R (s i) = Submodule.span R (⋃ i, s i)","typeReadable":"∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_9}\n  (s : ι → Set M), ⨆ i, Submodule.span R (s i) = Submodule.span R (⋃ i, s 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27))))) (s : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19) (n : N), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 f s (Submodule.span.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 (Singleton.singleton.{u_3, u_3} N (Set.{u_3} N) (Set.instSingletonSet.{u_3} N) n))) (Submodule.map.{u_1, u_1, u_2, u_4} R R M P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)) (DFunLike.coe.{max (max (succ u_2) (succ u_3)) (succ u_4), succ u_3, max (succ u_2) (succ u_4)} (LinearMap.{u_1, u_1, u_3, max u_4 u_2} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) N (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) M P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.13 (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_4} R R M P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 (LinearMap.module.{u_1, u_1, u_1, u_2, u_4} R R R M P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (SMulCommClass.symm.{u_1, u_1, u_4} R R P (SMulZeroClass.toSMul.{u_1, u_4} R P (AddZero.toZero.{u_4} P (AddZeroClass.toAddZero.{u_4} P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)) (DistribMulAction.toDistribSMul.{u_1, u_4} R P (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_4} R P (AddZero.toZero.{u_4} P (AddZeroClass.toAddZero.{u_4} P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)) (DistribMulAction.toDistribSMul.{u_1, u_4} R P (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27)))) (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27)))))) N (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : N) => LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) M P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27) (LinearMap.instFunLike.{u_1, u_1, u_3, max u_2 u_4} R R N (LinearMap.{u_1, u_1, u_2, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) M P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.13 (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_4} R R M P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 (LinearMap.module.{u_1, u_1, u_1, u_2, u_4} R R R M P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (SMulCommClass.symm.{u_1, u_1, u_4} R R P (SMulZeroClass.toSMul.{u_1, u_4} R P (AddZero.toZero.{u_4} P (AddZeroClass.toAddZero.{u_4} P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)) (DistribMulAction.toDistribSMul.{u_1, u_4} R P (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_1, u_4} R P (AddZero.toZero.{u_4} P (AddZeroClass.toAddZero.{u_4} P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R P (AddMonoid.toAddZeroClass.{u_4} P (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16)) (DistribMulAction.toDistribSMul.{u_1, u_4} R P (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27)))) (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27))))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)))) (LinearMap.flip.{u_1, u_1, u_1, u_1, u_2, u_3, u_4} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.27))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.665652973._hygCtx._hyg.7))) f) n) s)","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (s : Submodule R M) (n : N),\n  Submodule.map₂ f s (R ∙ n) = Submodule.map (f.flip n) s","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (s : Submodule R M) (n : N),\n  Submodule.map₂ f s (R ∙ n) = Submodule.map (f.flip n) 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(DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27))))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27) (LinearMap.instFunLike.{u_1, u_1, u_2, max u_3 u_4} R R M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.27)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7)))) f m) n))))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {f : M →ₗ[R] N →ₗ[R] P} {p : Submodule R M} {q : Submodule R N} {r : Submodule R P},\n  Submodule.map₂ f p q ≤ r ↔ ∀ m ∈ p, ∀ n ∈ q, (f m) n ∈ r","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {f : M →ₗ[R] N →ₗ[R] P} {p : Submodule R M} {q : Submodule R N} {r : Submodule R P},\n  Submodule.map₂ f p q ≤ r ↔ ∀ m ∈ p, ∀ n ∈ q, (f m) n ∈ r","typeReferences":[["smulCommClass_self"],["PartialOrder","toPreorder"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Preorder","toLE"],["SetLike","instMembership"],["LinearMap","module"],["CommSemiring","toSemiring"],["Submodule","instPartialOrder"],["LinearMap"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Submodule","setLike"],["Iff"],["LE","le"],["Submodule","map₂"]],"valueReferences":[["Submodule","map₂","_proof_2"],["smulCommClass_self"],["Subtype"],["PartialOrder","toPreorder"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["_private","Mathlib","Algebra","Module","Submodule","Bilinear",0,"Submodule","map₂_le","match_1_1"],["Submodule","map"],["Membership","mem"],["Submodule","map₂","_proof_1"],["AddCommMonoid","toAddMonoid"],["Subtype","val"],["DFunLike","coe"],["Submodule"],["Iff","intro"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["CompleteSemilatticeInf","toPartialOrder"],["DistribMulAction","toMulAction"],["Preorder","toLE"],["Submodule","map_le_iff_le_comap"],["Submodule","apply_mem_map₂"],["SetLike","instMembership"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap"],["CommSemiring","toCommMonoid"],["iSup_le"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Submodule","comap"],["Submodule","setLike"],["Iff","mpr"],["LE","le"],["Submodule","completeLattice"],["Submodule","map₂"],["Subtype","mk"],["CompleteLattice","toCompleteSemilatticeInf"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","Submodule","Bilinear",0,"Submodule","map₂_span_span","_simp_1_3"],"typeFallback":"forall {S : Type.{u_3}} {M : outParam.{succ (succ u_4)} Type.{u_4}} {inst._@.Mathlib.Algebra.Group.Submonoid.Defs.327041392._hygCtx._hyg.19 : Zero.{u_4} M} {inst._@.Mathlib.Algebra.Group.Submonoid.Defs.327041392._hygCtx._hyg.22 : SetLike.{u_3, u_4} S M} [self : ZeroMemClass.{u_3, u_4} S M inst._@.Mathlib.Algebra.Group.Submonoid.Defs.327041392._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Group.Submonoid.Defs.327041392._hygCtx._hyg.22] (s : S), Eq.{1} Prop (Membership.mem.{u_4, u_3} M S (SetLike.instMembership.{u_3, u_4} S M inst._@.Mathlib.Algebra.Group.Submonoid.Defs.327041392._hygCtx._hyg.22) s (OfNat.ofNat.{u_4} M 0 (Zero.toOfNat0.{u_4} M inst._@.Mathlib.Algebra.Group.Submonoid.Defs.327041392._hygCtx._hyg.19))) True","typeFull":"∀ {S : Type u_3} {M : outParam (Type u_4)} {inst : Zero M} {inst_1 : SetLike S M} [self : ZeroMemClass S M] (s : S),\n  (0 ∈ s) = True","typeReadable":"∀ {S : Type u_3} {M : outParam (Type u_4)} {inst : Zero M} {inst_1 : SetLike S M} [self : ZeroMemClass S M] (s : S),\n  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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27))))) -> (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.19) -> (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.23) -> (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27)","typeFull":"{R : Type u_1} →\n  {M : Type u_2} →\n    {N : Type u_3} →\n      {P : Type u_4} →\n        [inst : CommSemiring R] →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : AddCommMonoid N] →\n              [inst_3 : AddCommMonoid P] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R N] →\n                    [inst_6 : Module R P] → (M →ₗ[R] N →ₗ[R] P) → Submodule R M → Submodule R N → Submodule R P","typeReadable":"{R : Type u_1} →\n  {M : Type u_2} →\n    {N : Type u_3} →\n      {P : Type u_4} →\n        [inst : CommSemiring R] →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : AddCommMonoid N] →\n              [inst_3 : AddCommMonoid P] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R N] →\n                    [inst_6 : Module R P] → (M →ₗ[R] N →ₗ[R] P) → Submodule R M → Submodule R N → Submodule R P","typeReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submodule"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"]],"valueReferences":[["Submodule","map₂","_proof_2"],["Subtype"],["SetLike","instMembership"],["LinearMap","instFunLike"],["LinearMap","module"],["CommSemiring","toSemiring"],["Membership","mem"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Submodule","map"],["Submodule","map₂","_proof_1"],["LinearMap"],["Subtype","val"],["DFunLike","coe"],["Submodule"],["LinearMap","addCommMonoid"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Semiring","toNonAssocSemiring"],["iSup"],["RingHom","id"],["Submodule","setLike"],["ConditionallyCompletePartialOrderSup","toSupSet"],["Submodule","completeLattice"],["CompleteLattice","toConditionallyCompleteLattice"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_span_singleton_eq_map"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27))))) (m : M), Eq.{max (succ u_3) (succ u_4)} ((Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23) -> (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27)) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 f (Submodule.span.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.19 (Singleton.singleton.{u_2, u_2} M (Set.{u_2} M) (Set.instSingletonSet.{u_2} M) m))) (Submodule.map.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7)) (DFunLike.coe.{max (max (succ u_2) (succ u_3)) (succ u_4), succ u_2, max (succ u_3) (succ u_4)} (LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 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(CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.27)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3462318563._hygCtx._hyg.7)))) f m))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (m : M), Submodule.map₂ f (R ∙ m) = Submodule.map (f m)","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (m : M), Submodule.map₂ f (R ∙ m) = Submodule.map (f m)","typeReferences":[["smulCommClass_self"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Module"],["Singleton","singleton"],["CommSemiring"],["Submodule","map"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq"],["Set"],["LinearMap","module"],["CommSemiring","toSemiring"],["LinearMap"],["Set","instSingletonSet"],["CommSemiring","toCommMonoid"],["RingHomSurjective","ids"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Submodule","map₂"]],"valueReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Singleton","singleton"],["Submodule","map"],["Set","image2"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Submodule"],["Submodule","span_eq"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["funext"],["Eq","symm"],["DistribMulAction","toMulAction"],["Eq"],["Set"],["Submodule","map₂_span_span"],["CommSemiring","toSemiring"],["LinearMap","module"],["Set","instSingletonSet"],["LinearMap"],["CommSemiring","toCommMonoid"],["Set","image2_singleton_left"],["RingHomSurjective","ids"],["LinearMap","addCommMonoid"],["Submodule","map_span"],["Set","image"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Eq","refl"],["id"],["Submodule","map₂"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","Submodule","Bilinear",0,"Submodule","map₂_span_span","_simp_1_1"],"typeFallback":"forall {S : Type.{u_3}} {M : outParam.{succ (succ u_4)} Type.{u_4}} {inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.19 : Add.{u_4} M} {inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.22 : SetLike.{u_3, u_4} S M} [self : AddMemClass.{u_3, u_4} S M inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.22] {s : S} {a : M} {b : M}, (Membership.mem.{u_4, u_3} M S (SetLike.instMembership.{u_3, u_4} S M inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.22) s a) -> (Membership.mem.{u_4, u_3} M S (SetLike.instMembership.{u_3, u_4} S M inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.22) s b) -> (Eq.{1} Prop (Membership.mem.{u_4, u_3} M S (SetLike.instMembership.{u_3, u_4} S M inst._@.Mathlib.Algebra.Group.Subsemigroup.Defs.3909387615._hygCtx._hyg.22) s (HAdd.hAdd.{u_4, u_4, u_4} M M M (instHAdd.{u_4} M 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.43 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27))))) (g : LinearMap.{u_1, u_1, u_2, u_5} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) M M₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.43) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.19) (q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_5, u_3, u_4} R M₂ N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 f (Submodule.map.{u_1, u_1, u_2, u_5} R R M M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.43 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7)) g p) q) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (LinearMap.comp.{u_1, u_1, u_1, u_2, u_5, max u_4 u_3} R R R M M₂ (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.34 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.43 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.27)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2897324268._hygCtx._hyg.7))) (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R 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N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {M₂ : Type u_5} [inst_7 : AddCommMonoid M₂] [inst_8 : Module R M₂] (f : M₂ →ₗ[R] N →ₗ[R] P)\n  (g : M →ₗ[R] M₂) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f (Submodule.map g p) q = Submodule.map₂ (f ∘ₗ g) p q","typeReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["RingHomCompTriple","ids"],["Module"],["Submodule","map"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["RingHomSurjective","ids"],["Submodule"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Submodule","map₂"],["Eq"]],"valueReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["RingHomCompTriple","ids"],["Submodule","map"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["Submodule"],["congrArg"],["Submodule","map₂_map_right"],["Semiring","toNonAssocSemiring"],["LinearMap","flip"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq","symm"],["LinearMap","compl₂"],["Eq"],["LinearMap","module"],["CommSemiring","toSemiring"],["LinearMap"],["CommSemiring","toCommMonoid"],["RingHomSurjective","ids"],["LinearMap","addCommMonoid"],["Submodule","map₂_flip"],["Module","toDistribMulAction"],["Eq","refl"],["id"],["Submodule","map₂"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_le_map₂_left"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16] {f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27))))} {p₁ : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19} {p₂ : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19} {q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23}, (LE.le.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19) (Preorder.toLE.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19) (PartialOrder.toPreorder.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19) (Submodule.instPartialOrder.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19))) p₁ p₂) -> (LE.le.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27) (Preorder.toLE.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27) (PartialOrder.toPreorder.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27) (Submodule.instPartialOrder.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27))) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27 f p₁ q) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1261475627._hygCtx._hyg.27 f p₂ q))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {f : M →ₗ[R] N →ₗ[R] P} {p₁ p₂ : Submodule R M} {q : Submodule R N},\n  p₁ ≤ p₂ → Submodule.map₂ f p₁ q ≤ Submodule.map₂ f p₂ q","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {f : M →ₗ[R] N →ₗ[R] P} {p₁ p₂ : Submodule R M} {q : Submodule R N},\n  p₁ ≤ p₂ → Submodule.map₂ f p₁ q ≤ Submodule.map₂ f p₂ q","typeReferences":[["smulCommClass_self"],["PartialOrder","toPreorder"],["CommMonoid","toMonoid"],["Module"],["Submodule","instPartialOrder"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submodule"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LE","le"],["DistribMulAction","toMulAction"],["Submodule","map₂"],["Preorder","toLE"]],"valueReferences":[["le_refl"],["PartialOrder","toPreorder"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["Submodule","map₂_le_map₂"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {P : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16 : AddCommMonoid.{u_2} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27 : Module.{u_1, u_2} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16], SMulCommClass.{u_1, u_1, u_2} R R P (SemigroupAction.toSMul.{u_1, u_2} R P (Monoid.toSemigroup.{u_1} R (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7))) (MulAction.toSemigroupAction.{u_1, u_2} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7)) (DistribMulAction.toMulAction.{u_1, u_2} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_2} P 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_2} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.27))))","typeFull":"∀ {R : Type u_1} {P : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid P] [inst_2 : Module R P],\n  SMulCommClass R R P","typeReadable":"∀ {R : Type u_1} {P : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid P] [inst_2 : Module R P],\n  SMulCommClass R R P","typeReferences":[["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"],["SMulCommClass"],["AddCommMonoid"],["Module","toDistribMulAction"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]],"valueReferences":[["smulCommClass_self"],["Module","toDistribMulAction"],["CommMonoid","toMonoid"],["CommSemiring","toSemiring"],["DistribMulAction","toMulAction"],["AddCommMonoid","toAddMonoid"],["CommSemiring","toCommMonoid"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_iSup_left"],"typeFallback":"forall {ι : Sort.{uι}} {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27))))) (s : ι -> (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19)) (t : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27 f (iSup.{u_2, uι} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19) (Submodule.completeLattice.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19))))) (fun (i : ι) => s i)) t) (iSup.{u_4, uι} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) (CompleteLattice.toConditionallyCompleteLattice.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27) (Submodule.completeLattice.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27))))) (fun (i : ι) => Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.344583701._hygCtx._hyg.27 f (s i) t))","typeFull":"∀ {ι : Sort uι} {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M]\n  [inst_5 : Module R N] [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (s : ι → Submodule R M) (t : Submodule R N),\n  Submodule.map₂ f (⨆ i, s i) t = ⨆ i, Submodule.map₂ f (s i) t","typeReadable":"∀ {ι : Sort uι} {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M]\n  [inst_5 : Module R N] [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (s : ι → Submodule R M) (t : Submodule R N),\n  Submodule.map₂ f (⨆ i, s i) t = ⨆ i, Submodule.map₂ f (s i) 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7 : CommSemiring.{u_1} R], RingHomSurjective.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.970070980._hygCtx._hyg.7)))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R], RingHomSurjective (RingHom.id R)","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R], RingHomSurjective (RingHom.id 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16] {P₂ : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 : AddCommMonoid.{u_7} P₂] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51 : Module.{u_1, u_7} R P₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40] (f : LinearMap.{u_1, u_1, u_4, u_7} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) P P₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51) (g : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27))))) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.19) (q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23), Eq.{succ u_7} (Submodule.{u_1, u_7} R P₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51) (Submodule.map.{u_1, u_1, u_4, u_7} R R P P₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7)) f (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 g p q)) (Submodule.map₂.{u_1, u_2, u_3, u_7} R M N P₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51 (LinearMap.compr₂.{u_1, u_1, u_2, u_3, u_4, u_7} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7 R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) M N P P₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51 (smulCommClass_self.{u_1, u_7} R P₂ (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_7} R P₂ (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_7} P₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40) (Module.toDistribMulAction.{u_1, u_7} R P₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51))) (IsScalarTower.left.{u_1, u_7} R P₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) (DistribMulAction.toMulAction.{u_1, u_7} R P₂ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) (AddCommMonoid.toAddMonoid.{u_7} P₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40) (Module.toDistribMulAction.{u_1, u_7} R P₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.40 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.51))) (IsScalarTower.left.{u_1, u_4} R P (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) (DistribMulAction.toMulAction.{u_1, u_4} R P (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7))) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3058801722._hygCtx._hyg.27))) g f) p q)","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {P₂ : Type u_7} [inst_7 : AddCommMonoid P₂] [inst_8 : Module R P₂] (f : P →ₗ[R] P₂)\n  (g : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map f (Submodule.map₂ g p q) = Submodule.map₂ (g.compr₂ f) p q","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {P₂ : Type u_7} [inst_7 : AddCommMonoid P₂] [inst_8 : Module R P₂] (f : P →ₗ[R] P₂)\n  (g : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map f (Submodule.map₂ g p q) = Submodule.map₂ (g.compr₂ f) p 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.23 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.27)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7)))) f m) n) (SetLike.coe.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.19) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.19) p) (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.23) N (Submodule.setLike.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1071816252._hygCtx._hyg.23) q)))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f p q = Submodule.span R (Set.image2 (fun m n => (f m) n) ↑p ↑q)","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f p q = Submodule.span R (Set.image2 (fun m n => (f m) n) ↑p ↑q)","typeReferences":[["smulCommClass_self"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Module"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["Set","image2"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["CommSemiring","toCommMonoid"],["Submodule"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["SetLike","coe"],["Semiring","toNonAssocSemiring"],["Submodule","setLike"],["RingHom","id"],["DistribMulAction","toMulAction"],["Submodule","map₂"],["Eq"]],"valueReferences":[["smulCommClass_self"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Set","image2"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["congrArg"],["Submodule","span_eq"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq","symm"],["Eq"],["Submodule","map₂_span_span"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Eq","refl"],["id"],["Submodule","map₂"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_le_map₂_right"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16] {f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27))))} {p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.19} {q₁ : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23} {q₂ : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23}, (LE.le.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23) (Preorder.toLE.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23) (PartialOrder.toPreorder.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23) (Submodule.instPartialOrder.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23))) q₁ q₂) -> (LE.le.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27) (Preorder.toLE.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27) (PartialOrder.toPreorder.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27) (Submodule.instPartialOrder.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27))) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27 f p q₁) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.580558904._hygCtx._hyg.27 f p q₂))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {f : M →ₗ[R] N →ₗ[R] P} {p : Submodule R M} {q₁ q₂ : Submodule R N},\n  q₁ ≤ q₂ → Submodule.map₂ f p q₁ ≤ Submodule.map₂ f p q₂","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {f : M →ₗ[R] N →ₗ[R] P} {p : Submodule R M} {q₁ q₂ : Submodule R N},\n  q₁ ≤ q₂ → Submodule.map₂ f p q₁ ≤ Submodule.map₂ f p q₂","typeReferences":[["smulCommClass_self"],["PartialOrder","toPreorder"],["CommMonoid","toMonoid"],["Module"],["Submodule","instPartialOrder"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["Submodule"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LE","le"],["DistribMulAction","toMulAction"],["Submodule","map₂"],["Preorder","toLE"]],"valueReferences":[["le_refl"],["PartialOrder","toPreorder"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["Submodule","map₂_le_map₂"],["Submodule"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","Submodule","Bilinear",0,"Submodule","map₂_le","match_1_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7 : CommSemiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10] {p : Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19} (motive : (Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19)) p x)) -> Prop) (x._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx.135.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.143 : Subtype.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19)) p x)), (forall (m : M) (hm : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19)) p m), motive (Subtype.mk.{succ u_1} M (fun (x : M) => Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.19)) p x) m hm)) -> (motive x._@.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx.135.Mathlib.Algebra.Module.Submodule.Bilinear.1182562913._hygCtx._hyg.143)","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {p : Submodule R M} (motive : ↥p → Prop) (x : ↥p), (∀ (m : M) (hm : m ∈ p), motive ⟨m, hm⟩) → motive x","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  {p : Submodule R M} (motive : ↥p → Prop) (x : ↥p), (∀ (m : M) (hm : m ∈ p), motive ⟨m, hm⟩) → motive 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u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.27 f (Bot.bot.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.73548357._hygCtx._hyg.7) 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{R : Type.{u_1}} {M : Type.{u_4}} [inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 : Semiring.{u_1} R] [inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15 : Module.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12] {ι : Sort.{u_9}} (s : ι -> (Set.{u_4} M)), Eq.{succ u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (iSup.{u_4, u_9} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (CompleteLattice.toConditionallyCompleteLattice.{u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15) (Submodule.completeLattice.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15))))) (fun (i : ι) => Submodule.span.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15 (s i))) (Submodule.span.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.3630311781._hygCtx._hyg.15 (Set.iUnion.{u_4, u_9} M ι (fun (i : ι) => s i)))","typeFull":"∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_9}\n  (s : ι → Set M), ⨆ i, Submodule.span R (s i) = Submodule.span R (⋃ i, s i)","typeReadable":"∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_9}\n  (s : ι → Set M), ⨆ i, Submodule.span R (s i) = Submodule.span R (⋃ i, s i)","typeReferences":[["Module"],["Set"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Set","iUnion"],["Submodule","span"],["Submodule"],["AddCommMonoid"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["ConditionallyCompletePartialOrderSup","toSupSet"],["Submodule","completeLattice"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Semiring"]],"valueReferences":[["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["ConditionallyCompletePartialOrderSup","toSupSet"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Submodule","span_iUnion"],["Submodule","completeLattice"],["Eq","symm"],["Set","iUnion"],["Submodule","span"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_map_right"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16] {N₂ : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 : AddCommMonoid.{u_6} N₂] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47 : Module.{u_1, u_6} R N₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_6, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) N₂ P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27))))) (g : LinearMap.{u_1, u_1, u_3, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) N N₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.19) (q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.23), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_6, u_4} R M N₂ P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 f p (Submodule.map.{u_1, u_1, u_3, u_6} R R N N₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7)) g q)) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 (LinearMap.compl₂.{u_1, u_1, u_1, u_1, u_1, u_2, u_6, u_4, u_3} R R R R R M N₂ P N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.27 (RingHomCompTriple.ids.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7)))) (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3029185872._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R 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: M →ₗ[R] N₂ →ₗ[R] P)\n  (g : N →ₗ[R] N₂) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f p (Submodule.map g q) = Submodule.map₂ (f.compl₂ g) p q","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {N₂ : Type u_6} [inst_7 : AddCommMonoid N₂] [inst_8 : Module R N₂] (f : M →ₗ[R] N₂ →ₗ[R] P)\n  (g : N →ₗ[R] N₂) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f p (Submodule.map g q) = Submodule.map₂ (f.compl₂ g) p 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{ι : Sort.{uι}} {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27))))) (s : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.19) (t : ι -> (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23)), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27 f s (iSup.{u_3, uι} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23) (Submodule.completeLattice.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23))))) (fun (i : ι) => t i))) (iSup.{u_4, uι} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) (CompleteLattice.toConditionallyCompleteLattice.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27) (Submodule.completeLattice.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27))))) (fun (i : ι) => Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.62960218._hygCtx._hyg.27 f s (t i)))","typeFull":"∀ {ι : Sort uι} {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R]\n  [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M]\n  [inst_5 : Module R N] [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (s : Submodule R M) (t : ι → Submodule R N),\n  Submodule.map₂ f s (⨆ i, t i) = ⨆ i, Submodule.map₂ f s (t i)","typeReadable":"∀ {ι : Sort uι} {R : Type u_1} {M : Type u_2} 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{R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.7) 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.23))) q₁ q₂) -> (LE.le.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.27) (Preorder.toLE.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.27) (PartialOrder.toPreorder.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1387577082._hygCtx._hyg.16 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27))))) (p₁ : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (p₂ : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.23), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27 f (Max.max.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (Submodule.completeLattice.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19))))) p₁ p₂) q) (Max.max.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27) (SemilatticeSup.toMax.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27) (Lattice.toSemilatticeSup.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27) (ConditionallyCompleteLattice.toLattice.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27) (CompleteLattice.toConditionallyCompleteLattice.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27) (Submodule.completeLattice.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27))))) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.27 f p₁ q) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.13 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(p₁ p₂ : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f (p₁ ⊔ p₂) q = Submodule.map₂ f p₁ q ⊔ Submodule.map₂ f p₂ 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(R : Type.{u_1}) {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.11 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.14 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.11] {x : M}, Eq.{1} Prop (Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.14) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Lattice.26576063._hygCtx._hyg.11 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.27))))) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.19), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.4027884219._hygCtx._hyg.13 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AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M), Submodule.map₂ f p ⊥ = 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{R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16] {M₂ : Type.{u_5}} {N₂ : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.34 : AddCommMonoid.{u_5} M₂] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.37 : AddCommMonoid.{u_6} N₂] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.43 : Module.{u_1, u_5} R M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.34] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.47 : Module.{u_1, u_6} 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27))))) (g : LinearMap.{u_1, u_1, u_2, u_5} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7))) M M₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.43) (h : LinearMap.{u_1, u_1, u_3, u_6} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7))) N N₂ inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.47) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.19) (q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.23), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_5, u_6, u_4} R M₂ N₂ P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27 f (Submodule.map.{u_1, u_1, u_2, u_5} R R M M₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.43 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7)) g p) (Submodule.map.{u_1, u_1, u_3, u_6} R R N N₂ (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.47 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7))) (RingHomSurjective.ids.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7)) h q)) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27 (LinearMap.compl₁₂.{u_1, u_1, u_6, u_5, u_4, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) N₂ M₂ P M N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.43 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.47 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.23 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.1314993532._hygCtx._hyg.27))) f g h) p q)","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {M₂ : Type u_5} {N₂ : Type u_6} [inst_7 : AddCommMonoid M₂] [inst_8 : AddCommMonoid N₂]\n  [inst_9 : Module R M₂] [inst_10 : Module R N₂] (f : M₂ →ₗ[R] N₂ →ₗ[R] P) (g : M →ₗ[R] M₂) (h : N →ₗ[R] N₂)\n  (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f (Submodule.map g p) (Submodule.map h q) = Submodule.map₂ (f.compl₁₂ g h) p q","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] {M₂ : Type u_5} {N₂ : Type u_6} [inst_7 : AddCommMonoid M₂] [inst_8 : AddCommMonoid N₂]\n  [inst_9 : Module R M₂] [inst_10 : Module R N₂] (f : M₂ →ₗ[R] N₂ →ₗ[R] P) (g : M →ₗ[R] M₂) (h : N →ₗ[R] N₂)\n  (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f (Submodule.map g p) (Submodule.map h q) = Submodule.map₂ (f.compl₁₂ g h) p q","typeReferences":[["smulCommClass_self"],["LinearMap","compl₁₂"],["CommMonoid","toMonoid"],["Module"],["Submodule","map"],["CommSemiring"],["CommSemiring","toSemiring"],["LinearMap","module"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["CommSemiring","toCommMonoid"],["RingHomSurjective","ids"],["Submodule"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Submodule","map₂"],["Eq"]],"valueReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["RingHomCompTriple","ids"],["Submodule","map"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["Submodule"],["congrArg"],["Semiring","toNonAssocSemiring"],["Submodule","map₂_map_right"],["RingHom","id"],["DistribMulAction","toMulAction"],["LinearMap","compl₂"],["Eq"],["LinearMap","compl₁₂"],["Submodule","map₂_map_left"],["CommSemiring","toSemiring"],["LinearMap","module"],["LinearMap"],["CommSemiring","toCommMonoid"],["RingHomSurjective","ids"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Eq","refl"],["id"],["Submodule","map₂"],["Eq","mpr"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","Submodule","Bilinear",0,"Submodule","map₂_sup_left","match_1_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7 : CommSemiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19 : Module.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10] (p₁ : Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (p₂ : Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (_mn : M) (motive : (Exists.{succ u_1} M (fun (y : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₁ y) (Exists.{succ u_1} M (fun (z : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ z) (Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) y z) _mn))))) -> Prop) (x._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.101 : Exists.{succ u_1} M (fun (y : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₁ y) (Exists.{succ u_1} M (fun (z : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ z) (Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) y z) _mn))))), (forall (_m : M) (hm : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₁ _m) (_n : M) (hn : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ _n) (hmn : Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) _m _n) _mn), motive (Exists.intro.{succ u_1} M (fun (y : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₁ y) (Exists.{succ u_1} M (fun (z : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ z) (Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) y z) _mn)))) _m (And.intro (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₁ _m) (Exists.{succ u_1} M (fun (z : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ z) (Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) _m z) _mn))) hm (Exists.intro.{succ u_1} M (fun (z : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ z) (Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) _m z) _mn)) _n (And.intro (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19) M (Submodule.setLike.{u_2, u_1} R M (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.19)) p₂ _n) (Eq.{succ u_1} M (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.10)))) _m _n) _mn) hn hmn))))) -> (motive x._@.Mathlib.Algebra.Module.Submodule.Bilinear.3883696487._hygCtx._hyg.101)","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (p₁ p₂ : Submodule R M) (_mn : M) (motive : (∃ y ∈ p₁, ∃ z ∈ p₂, y + z = _mn) → Prop)\n  (x : ∃ y ∈ p₁, ∃ z ∈ p₂, y + z = _mn),\n  (∀ (_m : M) (hm : _m ∈ p₁) (_n : M) (hn : _n ∈ p₂) (hmn : _m + _n = _mn), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (p₁ p₂ : Submodule R M) (_mn : M) (motive : (∃ y ∈ p₁, ∃ z ∈ p₂, y + z = _mn) → Prop)\n  (x : ∃ y ∈ p₁, ∃ z ∈ p₂, y + z = _mn),\n  (∀ (_m : M) (hm : _m ∈ p₁) (_n : M) (hn : _n ∈ p₂) (hmn : _m + _n = _mn), motive ⋯) → motive x","typeReferences":[["Exists"],["SetLike","instMembership"],["Module"],["instHAdd"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["And"],["Exists","intro"],["Submodule"],["HAdd","hAdd"],["And","intro"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"]],"valueReferences":[["SetLike","instMembership"],["Exists"],["instHAdd"],["Membership","mem"],["CommSemiring","toSemiring"],["And"],["Exists","intro"],["Submodule"],["Exists","casesOn"],["HAdd","hAdd"],["And","intro"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","Submodule","Bilinear",0,"Submodule","map₂_span_span","_simp_1_2"],"typeFallback":"forall {R : Type.{u}} {M : Type.{v}} [inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 : AddCommMonoid.{v} M] {module_M : Module.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10} (p : Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) {x : M} (r : R), (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M)) p x) -> (Eq.{1} Prop (Membership.mem.{v, v} M (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) (SetLike.instMembership.{v, v} (Submodule.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M) M (Submodule.setLike.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M)) p (HSMul.hSMul.{u, v, v} R M M (instHSMul.{u, v} R M (SMulZeroClass.toSMul.{u, v} R M (AddZero.toZero.{v} M (AddZeroClass.toAddZero.{v} M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10)))) (DistribSMul.toSMulZeroClass.{u, v} R M (AddMonoid.toAddZeroClass.{v} M (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10)) (DistribMulAction.toDistribSMul.{u, v} R M (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{v} M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10) (Module.toDistribMulAction.{u, v} R M inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Defs.472439768._hygCtx._hyg.10 module_M))))) r x)) True)","typeFull":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (r : R), x ∈ p → (r • x ∈ p) = True","typeReadable":"∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)\n  {x : M} (r : R), x ∈ p → (r • x ∈ p) = True","typeReferences":[["SetLike","instMembership"],["True"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Submodule","smul_mem"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["eq_true"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Module","toDistribMulAction"],["Submodule","setLike"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","image2_subset_map₂"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27))))) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19) (q : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23), HasSubset.Subset.{u_4} (Set.{u_4} P) (Set.instHasSubset.{u_4} P) (Set.image2.{u_2, u_3, u_4} M N P (fun (m : M) (n : N) => DFunLike.coe.{max (succ u_3) (succ u_4), succ u_3, 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u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R 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N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27) (LinearMap.instFunLike.{u_1, u_1, u_2, max u_3 u_4} R R M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27)))) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7)))) f m) n) (SetLike.coe.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19) p) (SetLike.coe.{u_3, u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23) N (Submodule.setLike.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23) q)) (SetLike.coe.{u_4, u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27) P (Submodule.setLike.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3776732145._hygCtx._hyg.27 f p q))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Set.image2 (fun m n => (f m) n) ↑p ↑q ⊆ ↑(Submodule.map₂ f p q)","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Set.image2 (fun m n => (f m) n) ↑p ↑q ⊆ ↑(Submodule.map₂ f p q)","typeReferences":[["smulCommClass_self"],["CommMonoid","toMonoid"],["LinearMap","instFunLike"],["Module"],["CommSemiring"],["Set","image2"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Set"],["LinearMap","module"],["CommSemiring","toSemiring"],["LinearMap"],["CommSemiring","toCommMonoid"],["LinearMap","addCommMonoid"],["Set","instHasSubset"],["Module","toDistribMulAction"],["SetLike","coe"],["HasSubset","Subset"],["Submodule","setLike"],["Submodule","map₂"]],"valueReferences":[["smulCommClass_self"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq","ndrec"],["Eq"],["Submodule","apply_mem_map₂"],["Exists"],["Set"],["CommSemiring","toSemiring"],["LinearMap","module"],["And"],["LinearMap"],["CommSemiring","toCommMonoid"],["Set","instMembership"],["Exists","casesOn"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["Submodule","map₂"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_sup_right"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7))) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 (smulCommClass_self.{u_1, u_4} R P (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) (DistribMulAction.toMulAction.{u_1, u_4} R P (CommMonoid.toMonoid.{u_1} R (CommSemiring.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16) (Module.toDistribMulAction.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27))))) (p : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.19) (q₁ : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (q₂ : Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23), Eq.{succ u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 f p (Max.max.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SemilatticeSup.toMax.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (Lattice.toSemilatticeSup.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (ConditionallyCompleteLattice.toLattice.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (Submodule.completeLattice.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23))))) q₁ q₂)) (Max.max.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) (SemilatticeSup.toMax.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) (Lattice.toSemilatticeSup.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) (ConditionallyCompleteLattice.toLattice.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) (CompleteLattice.toConditionallyCompleteLattice.{u_4} (Submodule.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27) (Submodule.completeLattice.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27))))) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 f p q₁) (Submodule.map₂.{u_1, u_2, u_3, u_4} R M N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.27 f p q₂))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q₁ q₂ : Submodule R N),\n  Submodule.map₂ f p (q₁ ⊔ q₂) = Submodule.map₂ f p q₁ ⊔ Submodule.map₂ f p q₂","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q₁ q₂ : Submodule R N),\n  Submodule.map₂ f p (q₁ ⊔ q₂) = Submodule.map₂ f p q₁ ⊔ Submodule.map₂ f p 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{R : Type.{u_2}} {N : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7 : CommSemiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 : AddCommMonoid.{u_1} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23 : Module.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13] (q₁ : Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (q₂ : Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (_np : N) (motive : (Exists.{succ u_1} N (fun (y : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₁ y) (Exists.{succ u_1} N (fun (z : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ z) (Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) y z) _np))))) -> Prop) (x._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.101 : Exists.{succ u_1} N (fun (y : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₁ y) (Exists.{succ u_1} N (fun (z : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ z) (Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) y z) _np))))), (forall (_n : N) (hn : Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₁ _n) (_p : N) (hp : Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ _p) (hnp : Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) _n _p) _np), motive (Exists.intro.{succ u_1} N (fun (y : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₁ y) (Exists.{succ u_1} N (fun (z : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ z) (Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) y z) _np)))) _n (And.intro (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₁ _n) (Exists.{succ u_1} N (fun (z : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ z) (Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) _n z) _np))) hn (Exists.intro.{succ u_1} N (fun (z : N) => And (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ z) (Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) _n z) _np)) _p (And.intro (Membership.mem.{u_1, u_1} N (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23) N (Submodule.setLike.{u_2, u_1} R N (CommSemiring.toSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.23)) q₂ _p) (Eq.{succ u_1} N (HAdd.hAdd.{u_1, u_1, u_1} N N N (instHAdd.{u_1} N (AddCommMagma.toAdd.{u_1} N (AddCommSemigroup.toAddCommMagma.{u_1} N (AddCommMonoid.toAddCommSemigroup.{u_1} N inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.13)))) _n _p) _np) hp hnp))))) -> (motive x._@.Mathlib.Algebra.Module.Submodule.Bilinear.3555504422._hygCtx._hyg.101)","typeFull":"∀ {R : Type u_2} {N : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid N] [inst_2 : Module R N]\n  (q₁ q₂ : Submodule R N) (_np : N) (motive : (∃ y ∈ q₁, ∃ z ∈ q₂, y + z = _np) → Prop)\n  (x : ∃ y ∈ q₁, ∃ z ∈ q₂, y + z = _np),\n  (∀ (_n : N) (hn : _n ∈ q₁) (_p : N) (hp : _p ∈ q₂) (hnp : _n + _p = _np), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_2} {N : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid N] [inst_2 : Module R N]\n  (q₁ q₂ : Submodule R N) (_np : N) (motive : (∃ y ∈ q₁, ∃ z ∈ q₂, y + z = _np) → Prop)\n  (x : ∃ y ∈ q₁, ∃ z ∈ q₂, y + z = _np),\n  (∀ (_n : N) (hn : _n ∈ q₁) (_p : N) (hp : _p ∈ q₂) (hnp : _n + _p = _np), motive ⋯) → motive x","typeReferences":[["Exists"],["SetLike","instMembership"],["Module"],["instHAdd"],["CommSemiring"],["CommSemiring","toSemiring"],["Membership","mem"],["And"],["Exists","intro"],["Submodule"],["HAdd","hAdd"],["And","intro"],["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"]],"valueReferences":[["SetLike","instMembership"],["Exists"],["instHAdd"],["Membership","mem"],["CommSemiring","toSemiring"],["And"],["Exists","intro"],["Submodule"],["Exists","casesOn"],["HAdd","hAdd"],["And","intro"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Submodule","map₂_span_span"],"typeFallback":"forall (R : Type.{u_1}) {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.27 : 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inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2619055217._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R 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{R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} {P : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.10 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.13 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.16 : AddCommMonoid.{u_4} P] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.19 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.10] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.23 : Module.{u_1, u_3} R N (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.27 : Module.{u_1, u_4} R P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.16] (f : LinearMap.{u_1, u_1, u_2, max u_4 u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7))) M (LinearMap.{u_1, u_1, u_3, u_4} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7))) N P inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.10 (LinearMap.addCommMonoid.{u_1, u_1, u_3, u_4} R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7)))) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.19 (LinearMap.module.{u_1, u_1, u_1, u_3, u_4} R R R N P (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Bilinear.2160824120._hygCtx._hyg.23 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CommSemiring R] [inst_1 : AddCommMonoid M]\n  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]\n  [inst_6 : Module R P] (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N),\n  Submodule.map₂ f.flip q p = Submodule.map₂ f p 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