diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.ZMod.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.ZMod.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.ZMod.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["AddSubgroup","toZModSubmodule_symm"],"typeFallback":"forall (n : Nat) {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25 : Module.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9)], Eq.{succ u_1} ((Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25) -> (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9))) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (OrderIso.{u_1, 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inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25) (Submodule.instPartialOrder.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25))) (Preorder.toLE.{u_1} (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9)) (PartialOrder.toPreorder.{u_1} (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9)) (AddSubgroup.instPartialOrder.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9))))) (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25) => AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9)) (instFunLikeOrderIso.{u_1, u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.2125656325._hygCtx._hyg.25) (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M 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Submodule.toAddSubgroup","typeReferences":[["ZMod"],["CommRing","toCommSemiring"],["PartialOrder","toPreorder"],["Module"],["AddCommGroup","toAddGroup"],["Submodule","instPartialOrder"],["OrderIso"],["CommSemiring","toSemiring"],["AddSubgroup","instPartialOrder"],["ZMod","commRing"],["AddCommGroup"],["Submodule","toAddSubgroup"],["AddSubgroup","toZModSubmodule"],["DFunLike","coe"],["instFunLikeOrderIso"],["Submodule"],["CommRing","toRing"],["AddSubgroup"],["Nat"],["OrderIso","symm"],["AddCommGroup","toAddCommMonoid"],["Preorder","toLE"],["Eq"]],"valueReferences":[["rfl"],["ZMod"],["CommRing","toCommSemiring"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["OrderIso"],["AddSubgroup","instPartialOrder"],["ZMod","commRing"],["AddSubgroup","toZModSubmodule"],["DFunLike","coe"],["instFunLikeOrderIso"],["Submodule"],["AddSubgroup"],["OrderIso","symm"],["AddCommGroup","toAddCommMonoid"],["Preorder","toLE"]]},{"isProp":true,"kind":"theorem","name":["AddMonoidHom","toZModLinearMap","_proof_1"],"typeFallback":"forall {M : Type.{u_2}} {M₁ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.9 : AddCommGroup.{u_2} M] [inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.12 : AddCommGroup.{u_1} M₁] (f : AddMonoidHom.{u_2, u_1} M M₁ (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (SubNegMonoid.toAddMonoid.{u_2} M (AddGroup.toSubNegMonoid.{u_2} M (AddCommGroup.toAddGroup.{u_2} M inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.9))))) (AddZeroClass.toAddZero.{u_1} M₁ (AddMonoid.toAddZeroClass.{u_1} M₁ (SubNegMonoid.toAddMonoid.{u_1} M₁ (AddGroup.toSubNegMonoid.{u_1} M₁ (AddCommGroup.toAddGroup.{u_1} M₁ inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.12)))))) (x : M) (y : M), Eq.{succ u_1} M₁ (ZeroHom.toFun.{u_2, u_1} M M₁ (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (SubNegMonoid.toAddMonoid.{u_2} M (AddGroup.toSubNegMonoid.{u_2} M (AddCommGroup.toAddGroup.{u_2} M 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(SubNegMonoid.toAddMonoid.{u_2} M (AddGroup.toSubNegMonoid.{u_2} M (AddCommGroup.toAddGroup.{u_2} M inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.9))))))) x y)) (HAdd.hAdd.{u_1, u_1, u_1} M₁ M₁ M₁ (instHAdd.{u_1} M₁ (AddZero.toAdd.{u_1} M₁ (AddZeroClass.toAddZero.{u_1} M₁ (AddMonoid.toAddZeroClass.{u_1} M₁ (SubNegMonoid.toAddMonoid.{u_1} M₁ (AddGroup.toSubNegMonoid.{u_1} M₁ (AddCommGroup.toAddGroup.{u_1} M₁ inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.12))))))) (ZeroHom.toFun.{u_2, u_1} M M₁ (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (SubNegMonoid.toAddMonoid.{u_2} M (AddGroup.toSubNegMonoid.{u_2} M (AddCommGroup.toAddGroup.{u_2} M inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.9)))))) (AddZero.toZero.{u_1} M₁ (AddZeroClass.toAddZero.{u_1} M₁ (AddMonoid.toAddZeroClass.{u_1} M₁ (SubNegMonoid.toAddMonoid.{u_1} M₁ (AddGroup.toSubNegMonoid.{u_1} M₁ (AddCommGroup.toAddGroup.{u_1} M₁ 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(↑f).toFun y","typeReferences":[["instHAdd"],["AddCommGroup","toAddGroup"],["AddMonoidHom","toZeroHom"],["ZeroHom","toFun"],["AddCommGroup"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["AddMonoidHom"],["AddGroup","toSubNegMonoid"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddMonoidHom","map_add'"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toAddGroup"],["AddGroup","toSubNegMonoid"],["AddZeroClass","toAddZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["AddCommGroup","zmodModule","_proof_1"],"typeFallback":"forall (n._@.Mathlib.Algebra.Module.ZMod.3765744012._hygCtx._hyg.52 : Nat), NeZero.{0} Nat (Zero.ofOfNat0.{0} Nat (instOfNatNat 0)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n._@.Mathlib.Algebra.Module.ZMod.3765744012._hygCtx._hyg.52 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))","typeFull":"∀ (n : ℕ), NeZero (n + 1)","typeReadable":"∀ (n : 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inst._@.Mathlib.Algebra.Module.ZMod.543614420._hygCtx._hyg.57))) H' H)))))","typeFull":"��� {p : ℕ} {G : Type u_5} [inst : AddCommGroup G],\n  Nat.Prime p →\n    ∀ [inst_1 : Module (ZMod p) G] (H : Submodule (ZMod p) G) {k : ℕ},\n      k ≤ Nat.card ↥H → k ≠ 0 → ∃ H', Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' ∧ H' ≤ H","typeReadable":"∀ {p : ℕ} {G : Type u_5} [inst : AddCommGroup G],\n  Nat.Prime p →\n    ∀ [inst_1 : Module (ZMod p) G] (H : Submodule (ZMod p) G) {k : ℕ},\n      k ≤ Nat.card ↥H → k ≠ 0 → ∃ H', Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' ∧ H' ≤ 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S","typeReferences":[["ZMod"],["CommRing","toCommSemiring"],["PartialOrder","toPreorder"],["SetLike","instMembership"],["Module"],["AddCommGroup","toAddGroup"],["AddSubgroup","instSetLike"],["Submodule","instPartialOrder"],["OrderIso"],["CommSemiring","toSemiring"],["Membership","mem"],["AddSubgroup","instPartialOrder"],["ZMod","commRing"],["AddCommGroup"],["AddSubgroup","toZModSubmodule"],["DFunLike","coe"],["instFunLikeOrderIso"],["Submodule"],["AddSubgroup"],["Nat"],["Submodule","setLike"],["Iff"],["AddCommGroup","toAddCommMonoid"],["Preorder","toLE"]],"valueReferences":[["ZMod"],["CommRing","toCommSemiring"],["PartialOrder","toPreorder"],["SetLike","instMembership"],["AddCommGroup","toAddGroup"],["Submodule","instPartialOrder"],["Membership","mem"],["CommSemiring","toSemiring"],["OrderIso"],["AddSubgroup","instPartialOrder"],["ZMod","commRing"],["Iff","rfl"],["AddSubgroup","toZModSubmodule"],["DFunLike","coe"],["instFunLikeOrderIso"],["Submodule"],["AddSubgroup"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Preorder","toLE"]]},{"isProp":true,"kind":"theorem","name":["AddMonoidHom","toZModLinearMap","_proof_2"],"typeFallback":"forall (n : Nat) {M : Type.{u_2}} {M₁ : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.9 : AddCommGroup.{u_2} M] [inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.12 : AddCommGroup.{u_1} M₁] [inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.25 : Module.{0, u_2} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.9)] [inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.32 : Module.{0, u_1} (ZMod n) M₁ (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M₁ inst._@.Mathlib.Algebra.Module.ZMod.2862342879._hygCtx._hyg.12)] (f : AddMonoidHom.{u_2, u_1} M M₁ (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (SubNegMonoid.toAddMonoid.{u_2} M (AddGroup.toSubNegMonoid.{u_2} M 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x_2","typeReferences":[["ZMod"],["instAddNat"],["instHAdd"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["OfNat","ofNat"],["HAdd","hAdd"],["AddCommMonoid"],["Nat"],["Fin","val"],["AddMonoid","toNSMul"],["instOfNatNat"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["Nat"],["Fin","val"],["nsmul_add"],["instOfNatNat"],["instHAdd"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["AddSubgroup","toZModSubmodule"],"typeFallback":"forall (n : Nat) {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25 : Module.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9)], OrderIso.{u_1, u_1} 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inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 (RingHom.id.{0} (ZMod n) (Semiring.toNonAssocSemiring.{0} (ZMod n) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))))) (LinearMap.instFunLike.{0, 0, u_1, u_2} (ZMod n) (ZMod n) M M₁ (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9) (AddCommGroup.toAddCommMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 (RingHom.id.{0} (ZMod n) (Semiring.toNonAssocSemiring.{0} (ZMod n) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n)))))) (LinearMap.semilinearMapClass.{0, 0, u_1, u_2} (ZMod n) (ZMod n) M M₁ (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9) (AddCommGroup.toAddCommMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 (RingHom.id.{0} (ZMod n) (Semiring.toNonAssocSemiring.{0} (ZMod n) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n)))))))) g) (fun (f : AddMonoidHom.{u_1, u_2} M M₁ (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9))))) (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (SubNegMonoid.toAddMonoid.{u_2} M₁ (AddGroup.toSubNegMonoid.{u_2} M₁ (AddCommGroup.toAddGroup.{u_2} M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12)))))) => AddMonoidHom.toZModLinearMap.{u_1, u_2} n M M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 f)","typeFull":"∀ (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₁] [inst_2 : Module (ZMod n) M]\n  [inst_3 : Module (ZMod n) M₁], Function.LeftInverse (fun g => ↑g) fun f => AddMonoidHom.toZModLinearMap n f","typeReadable":"∀ (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₁] [inst_2 : Module (ZMod n) M]\n  [inst_3 : Module (ZMod n) M₁], Function.LeftInverse (fun g => ↑g) fun f => AddMonoidHom.toZModLinearMap n f","typeReferences":[["AddMonoidHomClass","toAddMonoidHom"],["RingHom"],["LinearMap","instFunLike"],["Module"],["AddCommGroup","toAddGroup"],["RingHom","instFunLike"],["DFunLike","coe"],["LinearMap","semilinearMapClass"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["ZMod"],["CommRing","toCommSemiring"],["AddMonoidHom","toZModLinearMap"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["ZMod","commRing"],["AddCommGroup"],["AddZeroClass","toAddZero"],["LinearMap"],["Module","toDistribMulAction"],["Nat"],["SubNegMonoid","toAddMonoid"],["Function","LeftInverse"],["AddCommGroup","toAddCommMonoid"],["SemilinearMapClass","distribMulActionSemiHomClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddMonoidHomClass","toAddMonoidHom"],["RingHom"],["LinearMap","instFunLike"],["AddCommGroup","toAddGroup"],["RingHom","instFunLike"],["DFunLike","coe"],["LinearMap","semilinearMapClass"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["AddMonoidHom"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["ZMod"],["CommRing","toCommSemiring"],["AddMonoidHom","toZModLinearMap"],["CommSemiring","toSemiring"],["ZMod","commRing"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Eq","refl"],["AddCommGroup","toAddCommMonoid"],["SemilinearMapClass","distribMulActionSemiHomClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","ZMod",0,"QuotientAddGroup","zmodModule","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3 : AddGroup.{u_1} α] {s : AddSubgroup.{u_1} α inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3} {C : (HasQuotient.Quotient.{u_1, u_1} α (AddSubgroup.{u_1} α inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3) (QuotientAddGroup.instHasQuotientAddSubgroup.{u_1} α inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3) s) -> Prop}, Eq.{1} Prop (forall (x : HasQuotient.Quotient.{u_1, u_1} α (AddSubgroup.{u_1} α inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3) (QuotientAddGroup.instHasQuotientAddSubgroup.{u_1} α inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3) s), C x) (forall (x : α), C (QuotientAddGroup.mk.{u_1} α inst._@.Mathlib.GroupTheory.Coset.Defs.3276294023._hygCtx._hyg.3 s x))","typeFull":"∀ {α : Type u_1} [inst : AddGroup α] {s : AddSubgroup α} {C : α ⧸ s → Prop}, (∀ (x : α ⧸ s), C x) = ∀ (x : α), C ↑x","typeReadable":"∀ {α : Type u_1} [inst : AddGroup α] {s : AddSubgroup α} {C : α ⧸ s → Prop}, (∀ (x : α ⧸ s), C x) = ∀ (x : α), C ↑x","typeReferences":[["AddSubgroup"],["HasQuotient","Quotient"],["QuotientAddGroup","instHasQuotientAddSubgroup"],["QuotientAddGroup","mk"],["AddGroup"],["Eq"]],"valueReferences":[["AddSubgroup"],["HasQuotient","Quotient"],["QuotientAddGroup","instHasQuotientAddSubgroup"],["QuotientAddGroup","mk"],["QuotientAddGroup","forall_mk"],["propext"]]},{"isProp":true,"kind":"theorem","name":["AddSubgroup","toZModSubmodule","_proof_4"],"typeFallback":"forall (n : Nat) {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25 : Module.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9)] {a._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.86 : AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9)} {b._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.87 : AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9)}, Iff (LE.le.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25) (Preorder.toLE.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25) 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(AddSubgroup.toAddSubmonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9) S) (AddSubgroup.toZModSubmodule._proof_3.{u_1} n M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25 S)) (Submodule.toAddSubgroup.{0, u_1} (ZMod n) M (CommRing.toRing.{0} (ZMod n) (ZMod.commRing n)) inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25) (AddSubgroup.toZModSubmodule._proof_1.{u_1} n M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25) (AddSubgroup.toZModSubmodule._proof_2.{u_1} n M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25)) a._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.86) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Equiv.{succ u_1, succ 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(AddSubgroup.toZModSubmodule._proof_2.{u_1} n M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25)) b._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.87)) (LE.le.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25) (Preorder.toLE.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.871633360._hygCtx._hyg.25) (PartialOrder.toPreorder.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing 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NeZero.{0} Nat (MulZeroClass.toZero.{0} Nat Nat.instMulZeroClass) n], Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (Zero.toOfNat0.{0} Nat (MulZeroClass.toZero.{0} Nat Nat.instMulZeroClass)))","typeFull":"∀ {n : ℕ} [NeZero n], n ≠ 0","typeReadable":"∀ {n : ℕ} [NeZero n], n ≠ 0","typeReferences":[["NeZero"],["Nat"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["Ne"],["Nat","instMulZeroClass"],["OfNat","ofNat"]],"valueReferences":[["Nat"],["MulZeroClass","toZero"],["Nat","instMulZeroClass"],["NeZero","ne"]]},{"isProp":true,"kind":"theorem","name":["ZMod","smul_mem"],"typeFallback":"forall {n : Nat} {M : Type.{u_1}} {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.9 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.25 : Module.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.9)] [inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.39 : SetLike.{u_4, u_1} S M] [inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.43 : AddSubgroupClass.{u_4, u_1} S M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.39] {x : M} {K : S}, (Membership.mem.{u_1, u_4} M S (SetLike.instMembership.{u_4, u_1} S M inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.39) K x) -> (forall (c : ZMod n), Membership.mem.{u_1, u_4} M S (SetLike.instMembership.{u_4, u_1} S M inst._@.Mathlib.Algebra.Module.ZMod.472439768._hygCtx._hyg.39) K (HSMul.hSMul.{0, u_1, u_1} (ZMod n) M M (instHSMul.{0, u_1} (ZMod n) M (SMulZeroClass.toSMul.{0, u_1} (ZMod n) M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M 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(AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.9)) (AddSubgroup.instPartialOrder.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.9)))) (Preorder.toLE.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.25) (PartialOrder.toPreorder.{u_1} (Submodule.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.25) (Submodule.instPartialOrder.{0, u_1} (ZMod n) M (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.25)))) (AddSubgroup.toZModSubmodule.{u_1} n M inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.3923086561._hygCtx._hyg.25) S)) S","typeFull":"∀ (n : ℕ) {M : Type u_1} [inst : AddCommGroup M] [inst_1 : Module (ZMod n) M] (S : AddSubgroup M),\n  ((AddSubgroup.toZModSubmodule n) S).toAddSubgroup = S","typeReadable":"∀ (n : ℕ) {M : Type u_1} [inst : AddCommGroup M] [inst_1 : Module (ZMod n) M] (S : AddSubgroup M),\n  ((AddSubgroup.toZModSubmodule n) S).toAddSubgroup = 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{n : Nat} {M : Type.{u_1}} {M₁ : Type.{u_2}} {F : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.9 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.12 : AddCommGroup.{u_2} M₁] [inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.15 : FunLike.{succ u_3, succ u_1, succ u_2} F M M₁] [inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.20 : AddMonoidHomClass.{u_3, u_1, u_2} F M M₁ (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.9))))) (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (SubNegMonoid.toAddMonoid.{u_2} M₁ (AddGroup.toSubNegMonoid.{u_2} M₁ (AddCommGroup.toAddGroup.{u_2} M₁ inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.12))))) inst._@.Mathlib.Algebra.Module.ZMod.4062404420._hygCtx._hyg.15] 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n) (ZMod.commRing n))))) M M₁ (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9) (AddCommGroup.toAddCommMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32) (LinearMap.instAdd.{0, 0, u_1, u_2} (ZMod n) (ZMod n) M M₁ (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9) (AddCommGroup.toAddCommMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 (RingHom.id.{0} (ZMod n) (Semiring.toNonAssocSemiring.{0} (ZMod n) (CommSemiring.toSemiring.{0} (ZMod n) (CommRing.toCommSemiring.{0} (ZMod n) (ZMod.commRing n))))))) (AddMonoidHom.toZModLinearMap.{u_1, u_2} n M M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 f₁) (AddMonoidHom.toZModLinearMap.{u_1, u_2} n M M₁ inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Module.ZMod.791860306._hygCtx._hyg.32 f₂))","typeFull":"∀ (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₁] [inst_2 : Module (ZMod n) M]\n  [inst_3 : Module (ZMod n) M₁] (f₁ f₂ : M →+ M₁),\n  AddMonoidHom.toZModLinearMap n (f₁ + f₂) = AddMonoidHom.toZModLinearMap n f₁ + AddMonoidHom.toZModLinearMap n f₂","typeReadable":"∀ (n : ℕ) {M : Type u_1} {M₁ : Type u_2} [inst : AddCommGroup M] [inst_1 : AddCommGroup M₁] [inst_2 : Module (ZMod n) M]\n  [inst_3 : Module (ZMod n) M₁] (f₁ f₂ : M →+ M₁),\n  AddMonoidHom.toZModLinearMap n (f₁ + f₂) = AddMonoidHom.toZModLinearMap n f₁ + AddMonoidHom.toZModLinearMap n f₂","typeReferences":[["ZMod"],["CommRing","toCommSemiring"],["AddMonoidHom","toZModLinearMap"],["AddMonoidHom","add"],["instHAdd"],["Module"],["AddCommGroup","toAddGroup"],["CommSemiring","toSemiring"],["ZMod","commRing"],["AddCommGroup"],["LinearMap"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Nat"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toAddMonoid"],["RingHom","id"],["AddMonoidHom"],["LinearMap","instAdd"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddMonoidHom","add"],["AddCommGroup","toAddGroup"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["AddMonoidHom"],["LinearMap","ext"],["LinearMap","instAdd"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["Eq"],["ZMod"],["CommRing","toCommSemiring"],["AddMonoidHom","toZModLinearMap"],["instHAdd"],["CommSemiring","toSemiring"],["ZMod","commRing"],["AddZeroClass","toAddZero"],["LinearMap"],["AddZero","toAdd"],["HAdd","hAdd"],["eq_self"],["SubNegMonoid","toAddMonoid"],["of_eq_true"],["AddCommGroup","toAddCommMonoid"],["AddMonoid","toAddZeroClass"]]}]