TensorProduct.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 8 curated AI-written annotations and 1204 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned FLT blueprint introduction

7 declarations, 0 contain sorry, 0 declared as axioms

Tensor Product

Material destined for Mathlib.

Imports (2)

Imported by 2 files

def ContinuousLinearEquiv.chooseBasisPiScalarRight Documented declaration: chooseBasisPiScalarRight line 24 source
Source documentation
The canonical continuous R-linear isomorphism M ⊗[R] V ≃ (ι → M) where V is a finite free R-module with basis indexed by ι, M is a commutative R-algebra, and M ⊗[R] V has the M-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology, ContinuousLinearEquiv, RingHom.id, Module.Free.ChooseBasisIndex
Proof uses: LinearEquiv, LinearEquiv.restrictScalars, LinearEquiv.chooseBasisPiScalarRight, ContinuousLinearEquiv.mk
54 instances, coercions and auxiliary constants left out.
```
noncomputable def ContinuousLinearEquiv.chooseBasisPiScalarRight (R M V : Type*)
    [CommRing M] [CommRing R] [Algebra R M]
    [TopologicalSpace M] [IsTopologicalRing M]
    [AddCommGroup V] [Module R V] [Module.Finite R V] [Module.Free R V]
    [TopologicalSpace (M ⊗[R] V)] [IsTopologicalAddGroup (M ⊗[R] V)]
    [IsModuleTopology M (M ⊗[R] V)] :
    (M ⊗[R] V) ≃L[R] (Module.Free.ChooseBasisIndex R V → M) := {
…
```
def ContinuousLinearEquiv.chooseBasisPiScalarRight' Documented declaration: chooseBasisPiScalarRight' line 41 source
Source documentation
The canonical continuous M-linear isomorphism M ⊗[R] V ≃ (ι → M) where V is a finite free R-module with basis indexed by ι, M is a commutative R-algebra, and M ⊗[R] V has the M-module topology.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology, ContinuousLinearEquiv, RingHom.id, Module.Free.ChooseBasisIndex
Proof uses: LinearEquiv, LinearEquiv.chooseBasisPiScalarRight, ContinuousLinearEquiv.mk
52 instances, coercions and auxiliary constants left out.
```
noncomputable def ContinuousLinearEquiv.chooseBasisPiScalarRight' (R M V : Type*)
    [CommRing M] [CommRing R] [Algebra R M]
    [TopologicalSpace M] [IsTopologicalRing M]
    [AddCommGroup V] [Module R V] [Module.Finite R V] [Module.Free R V]
    [TopologicalSpace (M ⊗[R] V)] [IsTopologicalAddGroup (M ⊗[R] V)]
    [IsModuleTopology M (M ⊗[R] V)] :
    (M ⊗[R] V) ≃L[M] (Module.Free.ChooseBasisIndex R V → M) := {
…
```
def ContinuousLinearMap.rTensor' The continuous R-linear map M ⊗[R] V → N ⊗[R] V induced by a continuous R-linear map M → N line 60 source
Source documentation
The continuous R-linear map M ⊗[R] V → N ⊗[R] V induced by a continuous R-linear map M → N.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, ContinuousLinearMap, RingHom.id, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology
Proof uses: LinearMap, LinearMap.rTensor, ContinuousLinearMap.mk
44 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
def rTensor' {R : Type*} {M N : Type*} (V : Type*)
    [CommRing M] [CommRing N] [CommRing R] [Algebra R M] [Algebra R N]
    [TopologicalSpace M] [TopologicalSpace N] [IsTopologicalRing M] [IsTopologicalRing N]
    (φ : M →L[R] N)
    [AddCommGroup V] [Module R V] [Module.Finite R V] [Module.Free R V]
    [TopologicalSpace (M ⊗[R] V)] [IsTopologicalAddGroup (M ⊗[R] V)] [IsModuleTopology M (M ⊗[R] V)]
    [TopologicalSpace (N ⊗[R] V)] [IsTopologicalAddGroup (N ⊗[R] V)]
    [IsModuleTopology N (N ⊗[R] V)] :
    (M ⊗[R] V) →L[R] (N ⊗[R] V) := {
…
```
lemma ContinuousLinearMap.rTensor'_id_apply line 102 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology, Eq, ContinuousLinearMap, RingHom.id, ContinuousLinearMap.rTensor', ContinuousLinearMap.id
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ContinuousLinearMap.mk.congr_simp, LinearMap.rTensor, LinearMap.id, LinearMap.rTensor_id, eq_self
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ContinuousLinearMap.rTensor'_id line 111 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology, Eq, ContinuousLinearMap, RingHom.id, ContinuousLinearMap.rTensor', ContinuousLinearMap.id
Proof uses: ContinuousLinearMap.ext, ContinuousLinearMap.rTensor'_id_apply
42 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma ContinuousLinearMap.rTensor'_comp_apply line 121 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, ContinuousLinearMap, RingHom.id, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology, Eq, ContinuousLinearMap.rTensor', ContinuousLinearMap.comp
Proof uses: of_eq_true, LinearMap, TensorProduct.map, LinearMap.comp, LinearMap.id, Eq.trans, True, congrArg, TensorProduct.map_map, eq_self
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ContinuousLinearMap.rTensor'_comp line 135 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, TopologicalSpace, IsTopologicalRing, ContinuousLinearMap, RingHom.id, AddCommGroup, Module, Module.Finite, Module.Free, TensorProduct, IsTopologicalAddGroup, IsModuleTopology, Eq, ContinuousLinearMap.rTensor', ContinuousLinearMap.comp
Proof uses: ContinuousLinearMap.ext, ContinuousLinearMap.rTensor'_comp_apply
44 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.