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r)","statement_sha256":"f7222d80328b10c9a944ecd2fb0b928935b52b96a172cd2c424f54338f424ff6","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-de70e30d8ca2f42509ec0d9eb6464224a1822979e523c4e486acefd2bd79d181","qualified_name":"AddMonoidHom.coe_mulRight","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":2102,"column":78,"line":66},"start":{"byte":2024,"column":0,"line":66}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_1} [inst : NonUnitalNonAssocSemiring.{u_1} R] (r : R),\n @Eq.{u_1 + 1} (R → R)\n (@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n (@AddMonoidHom.{u_1, u_1} R R\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst)))))\n R (fun x => R)\n (@AddMonoidHom.instFunLike.{u_1, u_1} R R\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst)))))\n (@AddMonoidHom.mulRight.{u_1} R inst r))\n fun x =>\n @HMul.hMul.{u_1, u_1, u_1} R R R\n (@instHMul.{u_1} R (@Distrib.toMul.{u_1} R (@NonUnitalNonAssocSemiring.toDistrib.{u_1} R inst))) x 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(@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))))\n f\n (@HMul.hMul.{u_1, u_1, u_1} R R R\n (@instHMul.{u_1} R (@Distrib.toMul.{u_1} R (@NonUnitalNonAssocSemiring.toDistrib.{u_1} R inst))) x y))\n (@HMul.hMul.{u_2, u_2, u_2} S S S\n (@instHMul.{u_2} S (@Distrib.toMul.{u_2} S (@NonUnitalNonAssocSemiring.toDistrib.{u_2} S inst_1)))\n (@DFunLike.coe.{max (u_1 + 1) (u_2 + 1), u_1 + 1, u_2 + 1}\n (@AddMonoidHom.{u_1, u_2} R S\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))))\n R (fun x => S)\n (@AddMonoidHom.instFunLike.{u_1, u_2} R S\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))))\n f x)\n (@DFunLike.coe.{max (u_1 + 1) (u_2 + 1), u_1 + 1, u_2 + 1}\n (@AddMonoidHom.{u_1, u_2} R S\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))))\n R (fun x => S)\n (@AddMonoidHom.instFunLike.{u_1, u_2} R S\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} 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(@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst)))\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))\n (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst)))\n (@AddMonoidHom.comp.{u_1, u_2, u_2} R S\n (@AddMonoidHom.{u_2, u_2} S S\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))))\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1))))\n (@AddZeroClass.toAddZero.{u_2}\n (@AddMonoidHom.{u_2, u_2} S S\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1)))))\n (@AddMonoid.toAddZeroClass.{u_2}\n (@AddMonoidHom.{u_2, u_2} S S\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} S inst_1))))\n (@AddZeroClass.toAddZero.{u_2} S\n (@AddMonoid.toAddZeroClass.{u_2} S\n (@AddCommMonoid.toAddMonoid.{u_2} S 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(@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))]\n {a : R},\n @Eq.{u_1 + 1} R\n (@Inv.inv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n a))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n (@Inv.inv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n a))","statement_sha256":"b11e9722a77fef750d3cd9e604276ebadd9c99c3b0becbcebc614376a7ed55bd","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e39a71f1509c9543f6ee08ad5e45ee29a95d354fe597e70e071d32d06d9b2dff","qualified_name":"inv_neg_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":11168,"column":70,"line":289},"start":{"byte":11098,"column":0,"line":289}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_1} [inst : DivisionMonoid.{u_1} R]\n [inst_1 :\n @HasDistribNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))],\n @Eq.{u_1 + 1} R\n (@Inv.inv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n (@OfNat.ofNat.{u_1} R (nat_lit 1)\n (@One.toOfNat1.{u_1} R\n (@InvOneClass.toOne.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))))))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n (@OfNat.ofNat.{u_1} R (nat_lit 1)\n (@One.toOfNat1.{u_1} R\n (@InvOneClass.toOne.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst))))))","statement_sha256":"b175a9da6f8e333f5bd62ffa8eddd66b552d6628c2367a784ec6687820e1daa8","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-82f81f3b6313f94cf1b5d8450f890e2531b1393207f7ec77cbedf7c605e8ff75","qualified_name":"isCancelMulZero_iff_isDomain_or_subsingleton","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":9213,"column":20,"line":239},"start":{"byte":8902,"column":0,"line":235}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (α : Type u_3) [inst : Semiring.{u_3} α],\n Iff\n (@IsCancelMulZero.{u_3} α (@Distrib.toMul.{u_3} α (@instDistribOfSemiring.{u_3} α inst))\n (@MulZeroClass.toZero.{u_3} α (@instMulZeroClassOfSemiring.{u_3} α inst)))\n (Or (@IsDomain.{u_3} α inst) (Subsingleton.{u_3 + 1} α))","statement_sha256":"722610c0e7083d5e751671a12c029cfc849b8a2d64a85fe6c2feaef83307a90b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ad6dc9234b8ab580e191243d47dcdfc267afc643776ead6609c660ced0c9112e","qualified_name":"isCancelMulZero_iff_noZeroDivisors","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":7537,"column":29,"line":202},"start":{"byte":7350,"column":0,"line":200}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_3} [inst : NonUnitalNonAssocRing.{u_3} R],\n Iff\n (@IsCancelMulZero.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n (@NoZeroDivisors.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))","statement_sha256":"1668ede091b3d190ece5113de03341521b7bcdec49249d848ac62f7566190fdf","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a4f6601c89e6feb91af0178cb1453eb0231ffa7fe1a315f4ce86e00648e1ce5d","qualified_name":"isDomain_iff_cancelMulZero_and_nontrivial","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":8900,"column":75,"line":233},"start":{"byte":8701,"column":0,"line":231}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (α : Type u_3) [inst : Semiring.{u_3} α],\n Iff (@IsDomain.{u_3} α inst)\n (And\n (@IsCancelMulZero.{u_3} α (@Distrib.toMul.{u_3} α (@instDistribOfSemiring.{u_3} α inst))\n (@MulZeroClass.toZero.{u_3} α (@instMulZeroClassOfSemiring.{u_3} α inst)))\n (Nontrivial.{u_3} α))","statement_sha256":"0f67c0c2bfaff8b614b3d7ae3ceddb28101ae8287386cbedf60f9121d210989d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-2a30b19b78cd5e75264e84d8c1098408a7ac71cda0b1fbc2d161fea4b8f8d315","qualified_name":"isDomain_iff_noZeroDivisors_and_nontrivial","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":9426,"column":88,"line":243},"start":{"byte":9215,"column":0,"line":241}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (α : Type u_3) [inst : Ring.{u_3} α],\n Iff (@IsDomain.{u_3} α (@Ring.toSemiring.{u_3} α inst))\n (And\n (@NoZeroDivisors.{u_3} α (@Distrib.toMul.{u_3} α (@instDistribOfSemiring.{u_3} α (@Ring.toSemiring.{u_3} α inst)))\n (@MulZeroClass.toZero.{u_3} α (@instMulZeroClassOfSemiring.{u_3} α (@Ring.toSemiring.{u_3} α inst))))\n (Nontrivial.{u_3} α))","statement_sha256":"e6c6b4dee1191997fa93f6056b84d2461c60c3cdd90866c60dfcbe274cd14f67","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f75efa1d445190da1eec5f56f98ac92360fca69795ccd6ee8ce025290c1aa110","qualified_name":"isLeftRegular_iff_right_eq_zero_of_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":6075,"column":83,"line":177},"start":{"byte":5846,"column":0,"line":175}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_3} [inst : NonUnitalNonAssocRing.{u_3} R] {r : R},\n Iff\n (@IsLeftRegular.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n r)\n (∀ (x : R),\n @Eq.{u_3 + 1} R\n (@HMul.hMul.{u_3, u_3, u_3} R R R\n (@instHMul.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n r x)\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))) →\n @Eq.{u_3 + 1} R x\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))))","statement_sha256":"5397f4b328c761ea6c2e47579055a996ec9264e08ce8018b3f78fbf781398e70","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e7ba3498ef1b16ae7d8ec946ac08101c6188da6911087c39f9bad30f3084de24","qualified_name":"isRegular_iff_eq_zero_of_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":6534,"column":100,"line":185},"start":{"byte":6309,"column":0,"line":183}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_3} [inst : NonUnitalNonAssocRing.{u_3} R] {r : R},\n Iff\n (@IsRegular.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n r)\n (And\n (∀ (x : R),\n @Eq.{u_3 + 1} R\n (@HMul.hMul.{u_3, u_3, u_3} R R R\n (@instHMul.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n r x)\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))) →\n @Eq.{u_3 + 1} R x\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))))\n (∀ (x : R),\n @Eq.{u_3 + 1} R\n (@HMul.hMul.{u_3, u_3, u_3} R R R\n (@instHMul.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n x r)\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))) →\n @Eq.{u_3 + 1} R x\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))))))","statement_sha256":"a9c2fa81a4cf2dcb375fb72342c33e6bcb8e6750c545559ccee01684d0bb567d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-5e8db16559c94d86bc109f21b6e1bcdfb31273ecaf928cf5ea13466defadc08b","qualified_name":"isRightRegular_iff_left_eq_zero_of_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":6307,"column":83,"line":181},"start":{"byte":6077,"column":0,"line":179}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_3} [inst : NonUnitalNonAssocRing.{u_3} R] {r : R},\n Iff\n (@IsRightRegular.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n r)\n (∀ (x : R),\n @Eq.{u_3 + 1} R\n (@HMul.hMul.{u_3, u_3, u_3} R R R\n (@instHMul.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n x r)\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))) →\n @Eq.{u_3 + 1} R x\n (@OfNat.ofNat.{u_3} R (nat_lit 0)\n (@Zero.toOfNat0.{u_3} R\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))))","statement_sha256":"92ca37db71e6eb02ff7b394434d987b9006717de00dccb9bd0df877b86b3342f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c15e939bb15233d160bb477b47627b6242167331726156491c954701c16e6e12","qualified_name":"isSelfInv_neg_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":11294,"column":53,"line":293},"start":{"byte":11170,"column":0,"line":291}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_1} [inst : DivisionMonoid.{u_1} R]\n [inst_1 :\n @HasDistribNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))]\n {a : R},\n Iff\n (@IsSelfInv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n a))\n (@IsSelfInv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n 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a))","statement_sha256":"1495a235dfd873c8f72bf6f68d838c31d22ab727bc2f03b7e813146a39b18838","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-8730d264a3ab5f20bf81ab93fb0355004f68cb954cd2591ece235fb2cdf7b4a8","qualified_name":"neg_div'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":10668,"column":63,"line":275},"start":{"byte":10605,"column":0,"line":275}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_1} [inst : DivisionMonoid.{u_1} R]\n [inst_1 :\n @HasDistribNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} 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(@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))]\n {a : R},\n @Eq.{u_1 + 1} R\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n (@Inv.inv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n a))\n (@Inv.inv.{u_1} R\n (@InvOneClass.toInv.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n 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α))","statement_sha256":"b47a8c83c3e7171888a87ab0ad27838b270f53bf43fad6cf55d0739d47c34856","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-65a9f052172a042e5bcb809bd7b29069e72c2ea440aa55709f276729a8868188","qualified_name":"noZeroDivisors_tfae","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":7348,"column":13,"line":198},"start":{"byte":6536,"column":0,"line":187}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_3} [inst : NonUnitalNonAssocRing.{u_3} R],\n (@List.cons.{0} Prop\n (@NoZeroDivisors.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n (@List.cons.{0} Prop\n (@IsLeftCancelMulZero.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n (@List.cons.{0} Prop\n (@IsRightCancelMulZero.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n (@List.cons.{0} Prop\n (@IsCancelMulZero.{u_3} R\n (@Distrib.toMul.{u_3} R\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst)))\n (@MulZeroClass.toZero.{u_3} R\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} R inst))))\n (@List.nil.{0} Prop))))).TFAE","statement_sha256":"b1344f9fb19921a92f7e3c8c6388707420f4de2820d5926832051cce794b62ff","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-107a01b4d2baa2fca58313a878fe10b52373eb52d0c3fddcb5ecf7534188669c","qualified_name":"one_div_neg_eq_neg_one_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":10199,"column":44,"line":263},"start":{"byte":9911,"column":0,"line":258}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {R : Type u_1} [inst : DivisionMonoid.{u_1} R]\n [inst_1 :\n @HasDistribNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))]\n (a : R),\n @Eq.{u_1 + 1} R\n (@HDiv.hDiv.{u_1, u_1, u_1} R R R\n (@instHDiv.{u_1} R (@DivInvMonoid.toDiv.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))\n (@OfNat.ofNat.{u_1} R (nat_lit 1)\n (@One.toOfNat1.{u_1} R\n (@InvOneClass.toOne.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n a))\n (@Neg.neg.{u_1} R\n (@InvolutiveNeg.toNeg.{u_1} R\n (@HasDistribNeg.toInvolutiveNeg.{u_1} R\n (@MulOne.toMul.{u_1} R\n (@MulOneClass.toMulOne.{u_1} R\n (@Monoid.toMulOneClass.{u_1} R\n (@DivInvMonoid.toMonoid.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))))\n inst_1))\n (@HDiv.hDiv.{u_1, u_1, u_1} R R R\n (@instHDiv.{u_1} R (@DivInvMonoid.toDiv.{u_1} R (@DivisionMonoid.toDivInvMonoid.{u_1} R inst)))\n (@OfNat.ofNat.{u_1} R (nat_lit 1)\n (@One.toOfNat1.{u_1} R\n (@InvOneClass.toOne.{u_1} R\n (@DivInvOneMonoid.toInvOneClass.{u_1} R (@DivisionMonoid.toDivInvOneMonoid.{u_1} R inst)))))\n 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a","statement_sha256":"bc1ce2700817319d191314194ed48d1580418e721617b07e7f7551f40d30b90b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ce1fc4035d8379b7d8f7b45fe15a609529acffd1557ed930a3bff243d4178d76","qualified_name":"vieta_formula_quadratic","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Ring.Basic","path":"Mathlib/Algebra/Ring/Basic.lean","range":{"end":{"byte":5348,"column":43,"line":157},"start":{"byte":4687,"column":0,"line":149}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_3} [inst : NonUnitalCommRing.{u_3} α] {b c x : α},\n @Eq.{u_3 + 1} α\n (@HAdd.hAdd.{u_3, u_3, u_3} α α α\n (@instHAdd.{u_3} α\n (@Distrib.toAdd.{u_3} 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(@instHSub.{u_3} α\n (@SubNegMonoid.toSub.{u_3} α\n (@AddGroup.toSubNegMonoid.{u_3} α\n (@AddCommGroup.toAddGroup.{u_3} α\n (@NonUnitalNonAssocRing.toAddCommGroup.{u_3} α\n (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} α\n (@NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} α inst)))))))\n (@HMul.hMul.{u_3, u_3, u_3} α α α\n (@instHMul.{u_3} α\n (@Distrib.toMul.{u_3} α\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} α\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} α\n (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} α\n (@NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} α inst))))))\n y y)\n (@HMul.hMul.{u_3, u_3, u_3} α α α\n (@instHMul.{u_3} α\n (@Distrib.toMul.{u_3} α\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} α\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} α\n (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} α\n (@NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} α inst))))))\n b y))\n c)\n (@OfNat.ofNat.{u_3} α (nat_lit 0)\n (@Zero.toOfNat0.{u_3} α\n (@MulZeroClass.toZero.{u_3} α\n (@NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} α\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} α\n (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} α\n (@NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} α inst))))))))\n (And\n (@Eq.{u_3 + 1} α\n (@HAdd.hAdd.{u_3, u_3, u_3} α α α\n (@instHAdd.{u_3} α\n (@Distrib.toAdd.{u_3} α\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} α\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} α\n (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} α\n (@NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_3} α inst))))))\n x y)\n b)\n (@Eq.{u_3 + 1} α\n (@HMul.hMul.{u_3, u_3, u_3} α α α\n (@instHMul.{u_3} α\n (@Distrib.toMul.{u_3} α\n (@NonUnitalNonAssocSemiring.toDistrib.{u_3} α\n (@NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} α\n (@NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_3} α\n 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