BaseChange.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

6 declarations, 0 contain sorry, 0 declared as axioms

Base Change

Material destined for Mathlib.

Imports (1)

Mathlib.RingTheory.Localization.BaseChange

Imported by 2 files

lemma IsLocalization.moduleLid_symm_apply line 29 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, Submonoid, Algebra, IsLocalization, AddCommMonoid, Module, IsScalarTower, Eq, TensorProduct, LinearEquiv, RingHom.id, LinearEquiv.symm, IsLocalization.moduleLid, TensorProduct.tmul, OfNat.ofNat
Proof uses: rfl
52 instances, coercions and auxiliary constants left out.
lemma IsLocalization.map_moduleTensorEquiv_tmul line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, Submonoid, Algebra, IsLocalization, AddCommMonoid, Module, IsScalarTower, Eq, TensorProduct, LinearEquiv, RingHom.id, IsLocalization.moduleTensorEquiv, TensorProduct.tmul
Proof uses: rfl
44 instances, coercions and auxiliary constants left out.
lemma IsLocalization.map_moduleTensorEquiv_symm_tmul line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, Submonoid, Algebra, IsLocalization, AddCommMonoid, Module, IsScalarTower, Eq, TensorProduct, LinearEquiv, RingHom.id, LinearEquiv.symm, IsLocalization.moduleTensorEquiv, TensorProduct.tmul
Proof uses: rfl
44 instances, coercions and auxiliary constants left out.
theorem IsLocalization.tensorProduct_ext line 56 source
standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, Submonoid, Algebra, IsLocalization, IsScalarTower, IsLocalizedModule, Algebra.linearMap, AddCommMonoid, Module, LinearMap, RingHom.id, TensorProduct, Eq, TensorProduct.tmul, RingHom, Algebra.algebraMap
Proof uses: TensorProduct.ext', Prod, Subtype, Membership.mem, HSMul.hSMul, Prod.snd, Prod.fst, Exists, IsLocalizedModule.surj, Prod.mk, Eq.mpr, id, congrArg, Eq.symm, propext, IsUnit.smul_left_cancel, IsLocalization.map_units, congr, Eq.trans, MulAction and 3 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
def IsLocalization.leftModuleTensorEquiv If A is a localization of R, tensoring two A-modules over A is the same as tensoring them over R line 70 source
Source documentation
If A is a localization of R, tensoring two A-modules over A is the same as tensoring them over R. This is IsLocalization.moduleTensorEquiv as an M₁-linear equivalence (more generally an M'-linear equivalence where M₁ is an M'-module).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Semiring, CommSemiring, Submonoid, Algebra, IsLocalization, AddCommMonoid, Module, SMulCommClass, IsScalarTower, LinearEquiv, RingHom.id, TensorProduct
Proof uses: IsLocalization.moduleTensorEquiv, LinearEquiv.mk, LinearMap.mk, LinearMap.toAddHom, LinearEquiv.invFun
41 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
noncomputable def leftModuleTensorEquiv {R : Type*} (M' : Type*)
    [Semiring M'] [CommSemiring R] (S : Submonoid R) (A : Type*) [CommSemiring A] [Algebra R A]
    [IsLocalization S A] (M₁ : Type*) (M₂ : Type*) [AddCommMonoid M₁] [AddCommMonoid M₂]
    [Module M' M₁] [Module R M₁] [Module R M₂] [Module A M₁] [Module A M₂]
    [SMulCommClass A M' M₁] [SMulCommClass R M' M₁] [IsScalarTower R A M₁]
    [IsScalarTower R A M₂] :
    M₁ ⊗[A] M₂ ≃ₗ[M'] M₁ ⊗[R] M₂ where
…
```
lemma IsLocalization.leftModuleTensorEquiv_restrictScalars_eq line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, Submonoid, Algebra, IsLocalization, AddCommMonoid, Module, IsScalarTower, Eq, LinearEquiv, RingHom.id, TensorProduct, LinearEquiv.restrictScalars, IsLocalization.leftModuleTensorEquiv, IsLocalization.moduleTensorEquiv
Proof uses: Eq.refl
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.