PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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.

Pullback is functorial up to canonical bundle isomorphism

Statement

For a smooth vector bundle $π : E \to M$ , there are canonical bundle isomorphisms

${id}_{M}^{*} E ≅ E, (f \circ g)^{*} E ≅ g^{*} (f^{*} E) .$

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ and smooth maps $g : P \to N$ and $f : N \to M$ .

[L1]

The pullback bundle is the fibre-product set with its smooth bundle structure (Pullback vector bundles as fibre products, The pullback fibre product is a smooth vector bundle).

Proof

technique · direct

1.1

For the identity map, define $I : {id}_{M}^{*} E \to E$ by $I (p, e) = e$ . This is well defined because $p = π (e)$ in the identity pullback, and its inverse is $e \mapsto (π (e), e)$ .

L1givenconstruct

2.1

An element of $g^{*} (f^{*} E)$ is a pair $(x, (g (x), e))$ with $f (g (x)) = π (e)$ . Send it to $(x, e) \in (f \circ g)^{*} E$ . The inverse is $(x, e) \mapsto (x, (g (x), e))$ . In the pulled-back bundle charts of [L1], both maps are the identity on the fibre coordinate, so they are smooth vector bundle isomorphisms.

L1step 1.1algebra∎

Depends on

Used by

Pullback of the tautological line bundle along the antipodal cover Example

Dependency tree · two levels

9 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)