LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

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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.

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The rational-defining relation is an equivalence relation

Statement

The relation $(a, b) \sim (c, d) ⟺ a d = c b$ on pairs of integers with nonzero second component (The rationals as equivalence classes of pairs of integers) is an equivalence relation.

Facts & Assumptions

Given: Pairs $(a, b), (c, d), (e, f)$ of integers with $b, d, f \neq 0$ .

[L1]

$Z$ is a commutative ring (The integers form a commutative ring).

[L2]

Multiplicative cancellation in $Z$ : $u d = v d$ with $d \neq 0$ implies $u = v$ (The integers have no zero divisors; multiplicative cancellation).

Proof

technique · direct

1.1

Reflexivity: $a b = b a$ , so $(a, b) \sim (a, b)$ .

1.2

Symmetry: if $a d = c b$ then $c b = a d$ , which is the defining equation for $(c, d) \sim (a, b)$ .

1.3

Suppose $(a, b) \sim (c, d)$ and $(c, d) \sim (e, f)$ , i.e. $a d = c b$ and $c f = e d$ .

given

2.1

Multiplying the first equation by $f$ and the second by $b$ : $a d f = c b f$ and $c f b = e d b$ .

step 1.3L1

3.1

Chaining: $(a f) d = a d f = c b f = c f b = e d b = (e b) d$ .

step 2.1L1

4.1

Cancelling the nonzero $d$ : $a f = e b$ , so $(a, b) \sim (e, f)$ ; the relation is transitive.

step 3.1L2

5.1

The relation is reflexive, symmetric, and transitive, hence an equivalence relation.

step 1.1step 1.2step 4.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The rationals as equivalence classes of pairs of integers.

Dependency tree · two levels

15 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

T. Tao, Analysis I, 3rd ed., §4.2 (standard reference, not scraped)
L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)