ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The kernel and image of the determinant homomorphism

Example

Assume ${A C}_{ω}$ and $n \geq 1$ . For real matrices,

$\det : G L_{n} (R) \to R^{\times}$

has kernel $S L_{n} (R)$ and image $R^{\times}$ . On $G L_{n}^{+} (R) = {A : \det A > 0}$ its image is $R_{> 0}$ . Hence the first-isomorphism factorization identifies the corresponding intrinsic quotients with these image Lie groups.

Facts & Assumptions

Given: ${A C}_{ω}$ and an integer $n \geq 1$ .

[F1]

The determinant is multiplicative and is given by its finite Leibniz formula. For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ . For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix.

[A1]

A smooth Lie-group homomorphism factors through its canonical immersed image as a surjective submersion followed by inclusion. The Axiom of Countable Choice ( ${A C}_{ω}$ ), First-isomorphism factorization for Lie group homomorphisms.

Verification

technique · compute kernel and image explicitly

1.1

Multiplicativity in [F1] and polynomiality make determinant a smooth Lie-group homomorphism. By definition, its identity fibre is ${A : \det A = 1} = S L_{n} (R)$ .

F1algebra

1.2

For each $r \in R^{\times}$ , the diagonal matrix $diag (r, 1, \dots, 1)$ is invertible and has determinant $r$ . Thus the image on $G L_{n} (R)$ is all of $R^{\times}$ . The same matrix lies in $G L_{n}^{+} (R)$ exactly when $r > 0$ , so the restricted image is $R_{> 0}$ .

F1algebra

2.1

Apply [A1]. It gives surjective submersions $G L_{n} (R) \to R^{\times}, G L_{n}^{+} (R) \to R_{> 0}$ with fibres the left cosets of $S L_{n} (R)$ , followed in each case by the evident inclusion of the image. Equivalently, the canonical intrinsic quotient by the kernel is isomorphic to the displayed image Lie group. For $n = 1$ these maps are the identity on $R^{\times}$ and its positive subgroup. The disconnected two-component image in the first case is intentional. Countable choice is used only through [A1].

A1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)