ModuleTopology.lean`

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33 declarations, 0 contain sorry, 0 declared as axioms

Module Topology

Material destined for Mathlib.

Imports (11)

Imported by 6 files

theorem ModuleTopology.isModuleTopology line 29 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Add, SMul, IsModuleTopology, moduleTopology
Proof uses: IsModuleTopology.mk, rfl
lemma ModuleTopology.iff should be in Mathlib line 35 source
Source comment
should be in Mathlib

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Add, SMul, TopologicalSpace, Iff, IsModuleTopology, Eq, moduleTopology
Proof uses: Iff.intro, IsModuleTopology.eq_moduleTopology', IsModuleTopology.mk
```
lemma ModuleTopology.iff (R M : Type*) [Add M] [SMul R M] [TopologicalSpace R]
    [τ : TopologicalSpace M] : IsModuleTopology R M ↔ τ = moduleTopology R M :=
…
```
theorem IsModuleTopology.Module.continuous_bilinear_of_finite_free line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, CommSemiring, AddCommMonoid, Module, IsModuleTopology, IsTopologicalSemiring, Module.Finite, Module.Free, LinearMap, RingHom.id, Continuous, Prod, Prod.fst, Prod.snd
Proof uses: Module.Free.ChooseBasisIndex, Fintype, Module.Basis, Module.Free.chooseBasis, LinearEquiv, Module.Basis.equivFun, LinearMap.comp, LinearEquiv.symm, IsModuleTopology.continuous_bilinear_of_pi_fintype, Eq, Function.comp, Prod.mk, funext, of_eq_true, Eq.trans, True, congrArg, congrFun', LinearEquiv.symm_apply_apply, eq_self and 8 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem IsModuleTopology.continuous_mul' line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, Ring, Algebra, Module.Finite, IsModuleTopology, Continuous, Prod, HMul.hMul, Prod.fst, Prod.snd
Proof uses: IsModuleTopology.continuous_bilinear_of_finite_left, LinearMap.mul
31 instances, coercions and auxiliary constants left out.
lemma IsModuleTopology.topologicalSemiring line 93 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, Ring, Algebra, Module.Finite, IsModuleTopology, IsTopologicalSemiring
Proof uses: IsTopologicalSemiring.mk, IsModuleTopology.toContinuousAdd, ContinuousMul.mk, IsModuleTopology.continuous_mul'
30 instances, coercions and auxiliary constants left out.
theorem IsModuleTopology.continuous_mul line 111 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, Ring, Algebra, Module.Finite, IsModuleTopology, Continuous, Prod, HMul.hMul, Prod.fst, Prod.snd
Proof uses: TensorProduct, moduleTopology, IsModuleTopology.mk, rfl, Eq.mpr, LinearMap, RingHom.id, LinearMap.mul, eq_of_heq, Eq, HEq, Eq.refl, HEq.refl, Eq.symm, funext, IsModuleTopology.continuous_bilinear_of_finite_left
50 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma IsModuleTopology.Module.topologicalRing line 117 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, Ring, Algebra, Module.Finite, IsModuleTopology
Proof uses: IsTopologicalRing.mk, IsTopologicalSemiring.mk, IsModuleTopology.toContinuousAdd, ContinuousMul.mk, IsModuleTopology.continuous_mul, ContinuousNeg.mk, IsModuleTopology.continuous_neg
36 instances, coercions and auxiliary constants left out.
lemma IsModuleTopology.iff_Continuous_algebraMap line 130 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, Algebra, Iff, IsTopologicalModule, Continuous, RingHom, Algebra.algebraMap
Proof uses: Iff.intro, continuous_algebraMap, Prod, HSMul.hSMul, Prod.fst, Prod.snd, Continuous.fun_smul, Continuous.comp', Continuous.fst, continuous_id', Continuous.snd, ContinuousSMul, ContinuousSMul.mk, Eq.mp, congrArg, funext, CommSemiring, AddCommMonoid, Module, IsScalarTower and 3 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem IsModuleTopology.isTopologicalModule making this into an instance causes timeouts in the BaseChange file :-/ line 145 source
Source comment
making this into an instance causes timeouts in the BaseChange file :-/

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, AddCommGroup, Module, IsModuleTopology, IsTopologicalModule
Proof uses: IsTopologicalModule.mk, ContinuousSMul.mk, Eq.rec, moduleTopology, Eq, Continuous, Prod, HSMul.hSMul, Prod.fst, Prod.snd, ContinuousSMul.continuous_smul, ModuleTopology.continuousSMul, Eq.symm, eq_moduleTopology, ContinuousAdd.mk, HAdd.hAdd, ContinuousAdd.continuous_add, ModuleTopology.continuousAdd
40 instances, coercions and auxiliary constants left out.
```
theorem isTopologicalModule
    (M : Type*) [AddCommGroup M] [TopologicalSpace M] [Module R M]
    [IsModuleTopology R M] : IsTopologicalModule R M where
…
```
Mentions: IsTopologicalModule
lemma IsModuleTopology._root_.Algebra.moduleTopology_le line 155 source
lemma IsModuleTopology._root_.moduleTopology.trans If S is an R-algebra, finite as an R-module, with the module topology, then the S-module topology on an S-module coincides with the R-module topology line 169 source
Source documentation
If S is an R-algebra, finite as an R-module, with the module topology, then the S-module topology on an S-module coincides with the R-module topology.
```
lemma _root_.moduleTopology.trans [IsTopologicalRing R] [Module.Finite R S] [IsModuleTopology R S] :
    moduleTopology R M = moduleTopology S M := by
…
```
lemma IsModuleTopology.trans line 194 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, Algebra, AddCommGroup, Module, IsScalarTower, IsTopologicalRing, Module.Finite, IsModuleTopology, Iff
Proof uses: of_eq_true, Eq.trans, Eq, moduleTopology, True, congr, congrArg, propext, ModuleTopology.iff, moduleTopology.trans, iff_self
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem IsModuleTopology.of_continuous_isOpenMap_algebraMap An R-algebra S has the R-module topology if the embedding R →+* S is continuous and open line 212 source
Source documentation
An R-algebra S has the R-module topology if the embedding R →+* S is continuous and open.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, Algebra, Continuous, RingHom, Algebra.algebraMap, IsOpenMap, IsModuleTopology
Proof uses: IsModuleTopology.mk, ContinuousSMul, continuousSMul_of_algebraMap, moduleTopology, id, Iff.mpr, LE.le, continuous_id_iff_le, moduleTopology_le, LinearMap, RingHom.id, Algebra.linearMap, IsModuleTopology.continuous_of_linearMap, ModuleTopology.continuousAdd, ModuleTopology.continuousSMul, Set, IsOpen, Continuous.isOpen_preimage, Set.range, Eq.rec and 31 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
theorem of_continuous_isOpenMap_algebraMap (hcont : Continuous (algebraMap R S))
    (hopen : IsOpenMap (algebraMap R S)) : IsModuleTopology R S where
…
```
theorem IsModuleTopology.of_isQuotientMap line 281 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module, IsModuleTopology, LinearMap, RingHom.id, Topology.IsQuotientMap
Proof uses: Eq.mpr, Eq, moduleTopology, id, congrArg, propext, ModuleTopology.iff, TopologicalSpace.coinduced, inferInstance, ModuleTopology.eq_coinduced_of_surjective, Topology.IsQuotientMap.surjective, Topology.IsCoinducing.eq_coinduced, Topology.IsQuotientMap.isCoinducing, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem IsModuleTopology.of_isOpenMap_surjective line 286 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module, IsModuleTopology, ContinuousAdd, ContinuousSMul, LinearMap, RingHom.id, IsOpenMap, Function.Surjective
Proof uses: IsModuleTopology.of_isQuotientMap, IsOpenQuotientMap.isQuotientMap, IsOpenQuotientMap.mk, IsModuleTopology.continuous_of_linearMap
38 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem IsModuleTopology.of_inverse line 291 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, TopologicalSpace, AddCommGroup, Module, IsModuleTopology, ContinuousAdd, ContinuousSMul, LinearMap, RingHom.id, Continuous, Function.LeftInverse
Proof uses: IsModuleTopology.of_isQuotientMap, Topology.IsQuotientMap.of_inverse, IsModuleTopology.continuous_of_linearMap
38 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def IsModuleTopology.continuousLinearEquiv Documented declaration: continuousLinearEquiv line 318 source
Source documentation
Given a linear isomorphism between two topological modules with the module topology, upgrades it to a continuous linear isomorphism using the fact that linear maps between modules with the module topology are automatically continuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Semiring, AddCommMonoid, Module, IsModuleTopology, LinearEquiv, RingHom.id, ContinuousLinearEquiv
Proof uses: ContinuousLinearEquiv.mk, LinearEquiv.mk, LinearMap.mk, AddHom.mk, LinearEquiv.invFun
37 instances, coercions and auxiliary constants left out.
```
def continuousLinearEquiv {A B R : Type*} [TopologicalSpace A]
    [TopologicalSpace B] [TopologicalSpace R] [Semiring R] [AddCommMonoid A] [AddCommMonoid B]
    [Module R A] [Module R B] [IsModuleTopology R A] [IsModuleTopology R B]
    (e : A ≃ₗ[R] B) :
    A ≃L[R] B where
…
```
def IsModuleTopology.continuousLinearEquivOfIsBiscalar A linear equivalence e : A ≃ₗ[S] B which also has scalars S', with A and B both having the S'-module topology, is automatically continuous line 336 source
Source documentation
A linear equivalence e : A ≃ₗ[S] B which also has scalars S', with A and B both having the S'-module topology, is automatically continuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Semiring, AddCommMonoid, Module, TopologicalSpace, IsModuleTopology, LinearEquiv, RingHom.id, IsBiscalar, ContinuousLinearEquiv
Proof uses: ContinuousLinearEquiv.mk
36 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
def continuousLinearEquivOfIsBiscalar {A B : Type*} (S' : Type*) {S : Type*}
    [Semiring S'] [Semiring S] [AddCommMonoid A] [AddCommMonoid B] [Module S A]
    [Module S B] [Module S' A] [Module S' B] [TopologicalSpace A] [TopologicalSpace B]
    [TopologicalSpace S'] [IsModuleTopology S' A] [IsModuleTopology S' B] (e : A ≃ₗ[S] B)
    [IsBiscalar S S' e] :
    A ≃L[S] B where
…
```
Mentions: IsBiscalar
def IsModuleTopology.continuousAlgEquivOfIsBiscalar An algebra isomorphism e : A →ₐ[S] B which also has scalars S', with A and B both having the S'-module topology, is automatically continuous line 356 source
Source documentation
An algebra isomorphism e : A →ₐ[S] B which also has scalars S', with A and B both having the S'-module topology, is automatically continuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, Algebra, TopologicalSpace, IsModuleTopology, AlgEquiv, IsBiscalar, AlgHom, ContinuousAlgEquiv
Proof uses: ContinuousAlgEquiv.mk
20 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
def continuousAlgEquivOfIsBiscalar {A B : Type*} (S' : Type*) {S : Type*}
    [CommSemiring A] [CommSemiring B] [CommSemiring S'] [CommSemiring S]
    [Algebra S A] [Algebra S B] [Algebra S' A] [Algebra S' B] [TopologicalSpace A]
    [TopologicalSpace B] [TopologicalSpace S'] [IsModuleTopology S' A]
    [IsModuleTopology S' B] (e : A ≃ₐ[S] B) [IsBiscalar S S' e.toAlgHom] :
    A ≃A[S] B where
…
```
Mentions: IsBiscalar
theorem IsModuleTopology.continuousAlgEquivOfIsBiscalar_apply line 375 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, Algebra, TopologicalSpace, IsModuleTopology, AlgEquiv, IsBiscalar, AlgHom, Eq, ContinuousAlgEquiv, IsModuleTopology.continuousAlgEquivOfIsBiscalar
Proof uses: rfl
23 instances, coercions and auxiliary constants left out.
def IsModuleTopology.continuousAlgEquivOfAlgEquiv An algebra isomorphism between two topological algebras over R with the R-module topology is automatically an algebra homeomorphism line 385 source
Source documentation
An algebra isomorphism between two topological algebras over R with the R-module topology is automatically an algebra homeomorphism.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, CommSemiring, Semiring, Algebra, IsModuleTopology, AlgEquiv, ContinuousAlgEquiv
Proof uses: ContinuousAlgEquiv.mk
8 instances, coercions and auxiliary constants left out.
```
def continuousAlgEquivOfAlgEquiv {A B R : Type*} [TopologicalSpace A]
    [TopologicalSpace B] [TopologicalSpace R] [CommSemiring R] [Semiring A] [Semiring B]
    [Algebra R A] [Algebra R B] [IsModuleTopology R A] [IsModuleTopology R B]
    (e : A ≃ₐ[R] B) :
    A ≃A[R] B where
…
```
theorem IsModuleTopology.t2Space A free module with the module topology over a T2Space ring is a T2Space line 400 source
Source documentation
A free module with the module topology over a T2Space ring is a T2Space.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Semiring, AddCommGroup, Module, Module.Free, TopologicalSpace, T2Space, ContinuousAdd, ContinuousMul, IsModuleTopology
Proof uses: IsTopologicalAddGroup, IsModuleTopology.isTopologicalAddGroup, Eq.mpr, IsClosed, Singleton.singleton, Set, OfNat.ofNat, id, Eq, congrArg, propext, IsTopologicalAddGroup.t2Space_iff_zero_closed, LinearMap, RingHom.id, Finsupp, Module.Free.ChooseBasisIndex, Module.Basis.repr, Module.Free.chooseBasis, AddMonoidHom, Finsupp.coeFnAddHom and 25 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem t2Space (R : Type*) {M : Type*} [Semiring R] [AddCommGroup M] [Module R M] [Module.Free R M]
    [TopologicalSpace R] [TopologicalSpace M] [T2Space R]
    [ContinuousAdd R] [ContinuousMul R] [IsModuleTopology R M]
    : T2Space M := by
…
```
theorem IsModuleTopology.t2Space' A vector space with the module topology over a T2Space ring is a T2Space line 421 source
Source documentation
A vector space with the module topology over a T2Space ring is a T2Space.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, AddCommGroup, Module, TopologicalSpace, T2Space, ContinuousAdd, ContinuousMul, IsModuleTopology
Proof uses: IsModuleTopology.t2Space
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem t2Space' {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
    [TopologicalSpace K] [TopologicalSpace V] [T2Space K]
    [ContinuousAdd K] [ContinuousMul K] [mt : IsModuleTopology K V]
    : T2Space V := by
…
```
instance IsModuleTopology._root_.Prod.instFinite_leftAlgebra A is a finite (A × B)-module line 450 source
Source documentation
A is a finite (A × B)-module.
```
instance _root_.Prod.instFinite_leftAlgebra : Module.Finite (A × B) A :=
…
```
instance IsModuleTopology._root_.Prod.instFinite_rightAlgebra B is a finite (A × B)-module line 455 source
Source documentation
B is a finite (A × B)-module.
```
instance _root_.Prod.instFinite_rightAlgebra : Module.Finite (A × B) B :=
…
```
instance IsModuleTopology.Prod.instLeftAlgebra A has the (A × B)-module topology line 463 source
Source documentation
A has the (A × B)-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, IsModuleTopology, Prod, Prod.leftAlgebra
Proof uses: IsModuleTopology.of_continuous_isOpenMap_algebraMap, continuous_fst, isOpenMap_fst
16 instances, coercions and auxiliary constants left out.
```
instance Prod.instLeftAlgebra : IsModuleTopology (A × B) A :=
…
```
instance IsModuleTopology.Prod.instRightAlgebra B has the (A × B)-module topology line 467 source
Source documentation
B has the (A × B)-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, IsModuleTopology, Prod, Prod.rightAlgebra
Proof uses: IsModuleTopology.of_continuous_isOpenMap_algebraMap, continuous_snd, isOpenMap_snd
16 instances, coercions and auxiliary constants left out.
```
instance Prod.instRightAlgebra : IsModuleTopology (A × B) B :=
…
```
instance IsModuleTopology.Prod.instLeftModule If M has the A-module topology, then it also has the (A × B)-module topology line 471 source
Source documentation
If M has the A-module topology, then it also has the (A × B)-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, TopologicalSpace, IsModuleTopology, IsTopologicalRing, Prod, Prod.leftModule
Proof uses: IsScalarTower, Prod.leftAlgebra, IsScalarTower.of_algebraMap_smul, Eq, HSMul.hSMul, RingHom, Algebra.algebraMap, rfl, Prod.mk, Eq.mpr, id, congrArg, propext, IsModuleTopology.trans, inferInstance
62 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
instance Prod.instLeftModule : IsModuleTopology (A × B) M := by
…
```
instance IsModuleTopology.Prod.instRightModule If N has the B-module topology, then it also has the (A × B)-module topology line 477 source
Source documentation
If N has the B-module topology, then it also has the (A × B)-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, TopologicalSpace, IsModuleTopology, IsTopologicalRing, Prod, Prod.rightModule
Proof uses: IsScalarTower, Prod.rightAlgebra, IsScalarTower.of_algebraMap_smul, Eq, HSMul.hSMul, RingHom, Algebra.algebraMap, rfl, Prod.mk, Eq.mpr, id, congrArg, propext, IsModuleTopology.trans, inferInstance
62 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
instance Prod.instRightModule : IsModuleTopology (A × B) N := by
…
```
instance IsModuleTopology.instProd' If M has the A-module topology and N has the B-module topology then M × N has the (A × B)-module topology line 484 source
Source documentation
If M has the A-module topology and N has the B-module topology then M × N has the (A × B)-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, TopologicalSpace, IsModuleTopology, IsTopologicalRing, Prod, Prod.leftModule, Prod.rightModule
Proof uses: inferInstance
53 instances, coercions and auxiliary constants left out.
```
instance instProd' : IsModuleTopology (A × B) (M × N) := inferInstance
…
```
theorem IsModuleTopology.locallyCompactSpaceOfFinite can't be an instance because typeclass inference can't find R line 494 source
Source comment
can't be an instance because typeclass inference can't find `R`

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, Ring, IsTopologicalRing, AddCommGroup, Module, IsModuleTopology, LocallyCompactSpace, Module.Finite
Proof uses: Nat, Exists, Fin, Eq, Submodule, Submodule.span, Set.range, Top.top, Module.Finite.exists_fin, IsOpenQuotientMap.locallyCompactSpace, LinearMap, RingHom.id, Fintype.linearCombination, AddMonoidHom.isOpenQuotientMap_of_isQuotientMap, RingHom, IsModuleTopology.isQuotientMap_of_surjective, Iff.mp, LinearMap.range, Function.Surjective, LinearMap.range_eq_top and 2 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem locallyCompactSpaceOfFinite [LocallyCompactSpace R] [Module.Finite R M] :
    LocallyCompactSpace M := by
…
```
def IsModuleTopology.Module.Basis.equivFunHomeo line 515 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Module, Module.Finite, TopologicalSpace, IsTopologicalRing, IsModuleTopology, ContinuousLinearEquiv, RingHom.id, Fin, Module.finrank
Proof uses: LinearEquiv, Module.Basis.equivFun, Module.finBasisOfFinrankEq, ContinuousLinearEquiv.mk
66 instances, coercions and auxiliary constants left out.
lemma IsModuleTopology._root_.Module.Finite.secondCountabletopology line 530 source