CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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.

$(- 1) (- 1) = 1$

Statement

In any field $F$ , the identity $(- 1) (- 1) = 1$ holds.

Facts & Assumptions

Given: A field $F$ with multiplicative identity $1$ .

[L1]

For all elements $a, b$ of a field, $(- a) (- b) = a b$ (Sign rules for products: $(- a) b = - (a b)$ and $(- a) (- b) = a b$ ).

[L2]

The element $1$ is the multiplicative identity, so $1 \cdot 1 = 1$ (field axiom).

Proof

technique · direct

1.1

Setting $a = b = 1$ in the sign rule gives $(- 1) (- 1) = 1 \cdot 1$ .

2.1

Since $1 \cdot 1 = 1$ , we conclude $(- 1) (- 1) = 1$ .

step 1.1L2∎

Depends on

Sign rules for products: $(-a)b = -(ab)$ and $(-a)(-b) = ab$

Used by

FALSE: every real number has a real square root False statement

Dependency tree · two levels

3 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

W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)
Janssen and Lindsey, Rings with Inquiry: Fields (standard reference, not scraped)

(−1)(−1)=1(−1)(−1)=1