Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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  • 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.
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Change of variables for an absolutely continuous map under an absolutely continuous composition hypothesis

Statement

Assume the Axioms of Countable Choice and Dependent Choice. Under the hypotheses of Chain rule for an indefinite integral after an absolutely continuous composition, ∫abf(g(x))g′(x) dx=F(g(b))−F(g(a)).

Facts & Assumptions

Given: Countable choice, dependent choice, and f,F,g as in the cited chain-rule lemma.

Proof

technique · direct
1.1

The lemma identifies f(g)g′ with (F∘g)′ almost everywhere and proves it integrable.

given
2.1

Apply Fundamental theorem of calculus for absolutely continuous functions to the AC function F∘g.

step 1.1
3.1

Its endpoint formula is exactly the displayed equality, including a=b.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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