PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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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.

A map into a product is smooth iff its components are smooth

Statement

Let $P$ , $M$ , and $N$ be smooth manifolds, let $F : P \to M \times N$ be a map, and write

$f : = π_{M} \circ F, g : = π_{N} \circ F,$

where $π_{M}$ and $π_{N}$ are the product projections. Then $F$ is smooth if and only if both component maps $f : P \to M$ and $g : P \to N$ are smooth.

Facts & Assumptions

Given: Smooth manifolds $P$ , $M$ , $N$ ; a map $F : P \to M \times N$ ; and its components $f : = π_{M} \circ F$ , $g : = π_{N} \circ F$ .

[F1]

The product $M \times N$ carries a canonical smooth structure whose smooth charts are represented by product charts $(V \times W, φ \times ψ)$ built from smooth charts $(V, φ)$ of $M$ and $(W, ψ)$ of $N$ (Products of smooth manifolds have a canonical product smooth structure).

[F2]

A map between smooth manifolds is smooth exactly when it is continuous and, in smooth charts, one coordinate representative is smooth near each point ( $C^{r}$ and smooth maps between smooth manifolds).

[L1]

A map into a product is continuous exactly when its two components are continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice).

[A1]

For open Euclidean sets $A \subseteq R^{a}$ , $B \subseteq R^{b}$ , and $U \subseteq R^{m}$ , a map $H : U \to A \times B$ is smooth if and only if its two components are smooth, because all coordinate partial derivatives of $H$ are exactly the coordinate partial derivatives of the component maps.

Proof

technique · direct

1.1

Assume $F$ is smooth, and fix $p \in P$ with charts as below. By [F2] it is continuous, so [L1] makes both components $f$ and $g$ continuous. Fix $p \in P$ and choose a smooth chart $(U, α)$ of $P$ at $p$ , smooth charts $(V, φ)$ of $M$ at $f (p)$ and $(W, ψ)$ of $N$ at $g (p)$ , and the product chart $(V \times W, φ \times ψ)$ of $M \times N$ at $F (p)$ from [F1].

givenF1F2L1choose

2.1

Conversely assume that $f$ and $g$ are smooth, and fix $p \in P$ with charts as below. Then [F2] makes them continuous, so [L1] makes $F$ continuous. Fix $p \in P$ and choose smooth charts $(U, α)$ of $P$ , $(V, φ)$ of $M$ , and $(W, ψ)$ of $N$ exactly as in step 1.1.

F1F2L1choose

3.1

In the charts chosen in steps 1.1 and 2.1, the representative of $F$ is the pair $((φ \times ψ) \circ F \circ α^{- 1}) (x) = (φ \circ f \circ α^{- 1} (x), ψ \circ g \circ α^{- 1} (x))$ . Under the hypothesis of step 1.1, the left-hand side is smooth, so [A1] makes the two component representatives smooth; since $p$ was arbitrary, $f$ and $g$ are smooth by [F2]. Under the hypothesis of step 2.1, the two component representatives are smooth by [F2], so [A1] makes the left-hand side smooth, and [F2] makes $F$ smooth at $p$ , hence everywhere.

F1F2A1step 1.1step 2.1

4.1

Step 3.1 proves both directions of the equivalence.

step 3.1∎

Depends on

Used by

Restrictions, corestrictions, and products of smooth maps are smooth Proposition

Dependency tree · two levels

24 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

Rob van der Vorst, Introduction to differentiable manifolds, §2 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.4 (standard reference, not scraped)