CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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 differential of a diffeomorphism is an isomorphism

If $F : M \to N$ is a diffeomorphism, then $d F_{p} : T_{p} M \to T_{F (p)} N$ is a linear isomorphism for every $p \in M$ .

Given: A diffeomorphism $F : M \to N$ and a point $p \in M$ .

[L1]

Differentials satisfy the chain rule for smooth compositions (The chain rule for differentials of smooth maps).

technique · direct

1.1L1given

Apply [L1] to $F^{- 1} \circ F = {id}_{M}$ and $F \circ F^{- 1} = {id}_{N}$ .

2.1step 1.1

The resulting identities show that $d (F^{- 1})_{F (p)}$ is both a left and right inverse of $d F_{p}$ .

3.1step 2.1∎

Hence $d F_{p}$ is a linear isomorphism.

Depends on