IntegralClosure.lean`

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14 declarations, 0 contain sorry, 0 declared as axioms

The integral closure as a type

We define IntegralClosure R A as a type synonym for the integral closure of R in A, equipped with the obvious commutative-ring and algebra structures.

Imports (4)

Imported by 2 files

def IntegralClosure IntegralClosure R A is the integral closure of R in A as a type line 30 source
Source documentation
IntegralClosure R A is the integral closure of R in A as a type. We don't use the subring directly to avoid horrendous timeouts.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra
Proof uses: Subtype, Membership.mem, Subalgebra, integralClosure
6 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
def IntegralClosure : Type _ := integralClosure R A
```
instance (instance at line 32) line 32 source
instance (instance at line 34) line 34 source
instance (instance at line 36) line 36 source
instance (instance at line 38) line 38 source
instance (instance at line 41) line 41 source
instance (instance at line 44) line 44 source
instance (instance at line 46) line 46 source
instance mulSemiringActionIntegralClosure line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, Monoid, MulSemiringAction, SMulCommClass, IntegralClosure
Proof uses: MulSemiringAction.mk, DistribMulAction.mk, MulAction.mk, SemigroupAction.mk, SMul.mk, Subtype.mk, Membership.mem, Subalgebra, integralClosure, HSMul.hSMul
55 instances, coercions and auxiliary constants left out.
instance smulCommClass_integralClosure line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, Monoid, MulSemiringAction, SMulCommClass, IntegralClosure
Proof uses: SMulCommClass.mk, Subtype.ext, Membership.mem, Subalgebra, integralClosure, HSMul.hSMul, SMulCommClass.smul_comm
55 instances, coercions and auxiliary constants left out.
lemma not_isField_integralClosure line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra, ValuationSubring, Ne, Top.top, Not, IsField, IntegralClosure, Subtype, Membership.mem
Proof uses: FaithfulSMul, inferInstance, Mathlib.Tactic.Contrapose.contrapose₄, Eq, IsField.toField, AlgHom, IsFractionRing.liftAlgHom, Algebra.ofId, FaithfulSMul.algebraMap_injective, Iff.mp, LE.le, top_le_iff, IsIntegrallyClosed, IsIntegral, isIntegral_algHom_iff, RingHom.injective, AlgHom.toRingHom, Algebra.IsIntegral.isIntegral, RingHom, Algebra.algebraMap and 3 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
instance isInvariant_integralClosure line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra, Group, MulSemiringAction, SMulCommClass, Algebra.IsInvariant, ValuationSubring, Subtype, Membership.mem, IntegralClosure
Proof uses: Algebra.IsInvariant.mk, Subalgebra, integralClosure, Eq, HSMul.hSMul, Exists, RingHom, Algebra.algebraMap, Subtype.mk, Algebra.IsInvariant.isInvariant, id, congrArg, IsIntegral, IsIntegrallyClosed, Iff.mp, IsIntegralClosure.isIntegral_iff, Eq.mp, propext, isIntegral_algebraMap_iff, Exists.intro and 1 more
108 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
instance continuousSMulDiscrete_integralClosure line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, Algebra, Group, MulSemiringAction, SMulCommClass, TopologicalSpace, ContinuousSMulDiscrete, IntegralClosure
Proof uses: ContinuousSMulDiscrete.mk, Eq.mpr, IsOpen, Set.ofPred, Eq, HSMul.hSMul, Membership.mem, Subalgebra, integralClosure, Subtype, id, congrArg, Set, funext, ContinuousSMulDiscrete.isOpen_smul_eq
63 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
instance (instance at line 103) line 103 source