Matrix.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

1 declarations, 0 contain sorry, 0 declared as axioms

Matrix

Material destined for Mathlib.

Imports (3)

Imported by 2 files

theorem DenseRange.exists_matrix_det_ne_zero Let A : X → Matrix n n R be a parameterisation of a matrix ring with entries in a seminormed commutative ring R by a metric space X and let f : Y → X be dense line 24 source
Source documentation
Let A : X → Matrix n n R be a parameterisation of a matrix ring with entries in a seminormed commutative ring R by a metric space X and let f : Y → X be dense. If the identity matrix is in the parameterisation A, then there exists a y : Y such that A (f y) has non-zero determinant.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Fintype, DecidableEq, SeminormedCommRing, NormOneClass, IsTopologicalRing, Matrix, Continuous, DenseRange, Eq, OfNat.ofNat, Exists, Ne, Matrix.det
Proof uses: Set.range, Membership.mem, Set, Function.comp, Set.codRestrict, DenseRange.codRestrict_comp, Continuous.matrix_det, LT.lt, Real, Dist.dist, Subtype.mk, Eq.substr, Exists.intro, rfl, HDiv.hDiv, Nat, Eq.mp, Eq.trans, GT.gt, forall_congr and 86 more
131 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem DenseRange.exists_matrix_det_ne_zero {X Y n R : Type*} [PseudoMetricSpace X]
    [Fintype n] [DecidableEq n] [SeminormedCommRing R] [NormOneClass R]
    [IsTopologicalRing R] {A : X → Matrix n n R} {f : Y → X} (hA : Continuous A) (hf : DenseRange f)
    {b : X} (hb : A b = 1) :
    ∃ y, (A (f y)).det ≠ 0 := by
…
```