Quotient.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Pinned FLT blueprint introduction

2 declarations, 0 contain sorry, 0 declared as axioms

Quotient

Material destined for Mathlib.

Imports (2)

Imported by 2 files

def Submodule.Quotient.continuousLinearEquiv A continuous R-linear equivalence e : G ≃L[R] H sending the submodule G' ≤ G onto H' ≤ H descends to a continuous R-linear equivalence G ⧸ G' ≃L[R] H ⧸ H' line 21 source
Source documentation
A continuous R-linear equivalence e : G ≃L[R] H sending the submodule G' ≤ G onto H' ≤ H descends to a continuous R-linear equivalence G ⧸ G' ≃L[R] H ⧸ H'.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, AddCommGroup, Module, TopologicalSpace, Submodule, ContinuousLinearEquiv, RingHom.id, Eq, Submodule.map, HasQuotient.Quotient
Proof uses: LinearEquiv, Submodule.Quotient.equiv, ContinuousLinearEquiv.mk
25 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
def Submodule.Quotient.continuousLinearEquiv {R : Type*} [Ring R] (G H : Type*) [AddCommGroup G]
    [Module R G] [AddCommGroup H] [Module R H] [TopologicalSpace G] [TopologicalSpace H]
    (G' : Submodule R G) (H' : Submodule R H) (e : G ≃L[R] H)
    (h : Submodule.map e.toLinearMap G' = H') :
    (G ⧸ G') ≃L[R] (H ⧸ H') where
…
```
def Submodule.quotientPiContinuousLinearEquiv For a finite family of topological R-modules G i and submodules p i ≤ G i, the canonical continuous R-linear equivalence (∏ i, G i) ⧸ ∏ i, p i ≃L[R] ∏ i, G i ⧸ p i line 39 source
Source documentation
For a finite family of topological R-modules G i and submodules p i ≤ G i, the canonical continuous R-linear equivalence (∏ i, G i) ⧸ ∏ i, p i ≃L[R] ∏ i, G i ⧸ p i.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommGroup, Module, TopologicalSpace, IsTopologicalAddGroup, Fintype, DecidableEq, Submodule, ContinuousLinearEquiv, RingHom.id, HasQuotient.Quotient, Submodule.pi, Set.univ
Proof uses: LinearEquiv, Submodule.quotientPi, ContinuousLinearEquiv.mk
32 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
def Submodule.quotientPiContinuousLinearEquiv {R ι : Type*} [CommRing R] {G : ι → Type*}
    [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] [(i : ι) → TopologicalSpace (G i)]
    [(i : ι) → IsTopologicalAddGroup (G i)] [Fintype ι] [DecidableEq ι]
    (p : (i : ι) → Submodule R (G i)) :
    (((i : ι) → G i) ⧸ Submodule.pi Set.univ p) ≃L[R] ((i : ι) → G i ⧸ p i) where
…
```