Basic.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

3 declarations, 0 contain sorry, 0 declared as axioms

The internal hom of continuous representations

This file constructs ContRepresentation.linHom ρ2 ρ3, the continuous representation of G on M2 →L[k] M3 by conjugation, g • φ = ρ3 g ∘L φ ∘L ρ2 g⁻¹, where M2 →L[k] M3 carries the topology induced from the compact-open topology on C(M2, M3).

Material destined for Mathlib.RepresentationTheory.Continuous.Basic.

Imports (3)

Imported by 3 files

def ContRepresentation.linHom Documented declaration: linHom line 42 source
Source documentation
The continuous representation of G on M2 →L[k] M3 by conjugation, g • φ = ρ3 g ∘L φ ∘L ρ2 g⁻¹, where M2 →L[k] M3 carries the topology induced from the compact-open topology on C(M2, M3).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, Group, AddCommGroup, Module, IsTopologicalAddGroup, ContinuousSMul, ContRepresentation, ContinuousLinearMap, RingHom.id
Proof uses: ContRepresentation.ofMonoidHom, MonoidHom.mk, OneHom.mk, ContinuousLinearMap.mk, LinearMap.mk, AddHom.mk, ContinuousLinearMap.comp, Inv.inv
78 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
def linHom : ContRepresentation k G (M2 →L[k] M3) where
…
```
lemma ContRepresentation.linHom_apply line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, Group, AddCommGroup, Module, IsTopologicalAddGroup, ContinuousSMul, ContRepresentation, ContinuousLinearMap, RingHom.id, Eq, ContRepresentation.linHom, ContinuousLinearMap.comp, Inv.inv
Proof uses: rfl
71 instances, coercions and auxiliary constants left out.
lemma ContRepresentation.continuous_pair_of_discrete When the source module of an intertwining map into an internal hom is discrete, its uncurried pairing is automatically jointly continuous line 74 source
Source documentation
When the source module of an intertwining map into an internal hom is discrete, its uncurried pairing is automatically jointly continuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, Group, AddCommGroup, Module, IsTopologicalAddGroup, ContinuousSMul, ContRepresentation, DiscreteTopology, ContIntertwiningMap, ContinuousLinearMap, RingHom.id, ContRepresentation.linHom, Continuous, Prod, Prod.fst, Prod.snd
Proof uses: Continuous.comp, Prod.swap, continuous_of_discreteTopology_snd, ContinuousLinearMap.continuous, continuous_swap
68 instances, coercions and auxiliary constants left out.
```
lemma continuous_pair_of_discrete [DiscreteTopology M1] (f : ρ1 →ⁱL linHom ρ2 ρ3) :
    Continuous fun p : M1 × M2 ↦ f p.1 p.2 :=
…
```
Mentions: ContRepresentation.linHom