Monoid.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

2 declarations, 0 contain sorry, 0 declared as axioms

Monoid

Material destined for Mathlib.

Imports (2)

Imported by 1 file

FLT.lean

theorem induced_continuous_smul note: use convert not exact to ensure typeclass inference doesn't try to find topology on B line 35 source
Source comment
note: use convert not exact to ensure typeclass inference doesn't try to find topology on B

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, SMul, ContinuousSMul, MulActionHom, Continuous, TopologicalSpace.induced
Proof uses: Eq.mpr, Eq.refl, Topology.IsInducing.continuousSMul, Topology.IsInducing.induced, MulActionSemiHomClass.map_smulₛₗ
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem induced_continuous_smul [τA : TopologicalSpace A] [ContinuousSMul R A] (g : B →ₑ[f] A)
    (hf : Continuous f) : @ContinuousSMul S B _ _ (TopologicalSpace.induced g τA) := by
…
```
theorem induced_continuous_mul line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommMonoid, TopologicalSpace, ContinuousMul, MonoidHom, TopologicalSpace.induced
Proof uses: Eq.mpr, Eq.refl, Topology.IsInducing.continuousMul, Topology.IsInducing.induced
14 instances, coercions and auxiliary constants left out.

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