LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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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A finite product of nonzero reals is nonzero

Statement

Let $n \in N$ and let $a_{0}, \dots, a_{n - 1} \in R$ all be nonzero. Then the finite product $\prod_{k < n} a_{k}$ is nonzero. For $n = 0$ this is the empty product $1$ .

Facts & Assumptions

Given: A natural number $n$ and a finite list of nonzero real factors.

[F1]

Finite sums and finite products, by recursion and Laws of finite sums and finite products: $\prod_{k < 0} a_{k} = 1$ and $\prod_{k < n + 1} a_{k} = (\prod_{k < n} a_{k}) a_{n}$ .

[F2]

The principle of mathematical induction: a property true at zero and preserved by successors holds for every natural number.

[F3]

The reals form a field and Field: $R$ is a field and its identities satisfy $1 \neq 0$ .

[F4]

A field has no zero divisors: $a b = 0 \Rightarrow a = 0$ or $b = 0$ : if a product of two field elements is zero, at least one factor is zero.

Proof

technique · induction

1.1basegivenF1F3

For $n = 0$ , the empty product is $1$ , which is nonzero by [F3].

1.2givenF1F4ihalgebra

Suppose the prefix product $\prod_{k < n} a_{k}$ is nonzero and the next factor $a_{n}$ is nonzero. By [F1] the extended product is $(\prod_{k < n} a_{k}) a_{n}$ ; [F4] implies it cannot be zero, since both factors are nonzero.

2.1step 1.1step 1.2F2discharge-induction∎

Induction [F2] applied to steps 1.1 and 1.2 proves the claim for every finite list of nonzero real factors.

Depends on

Used by

Dependency tree · two levels

27 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

Mark Embree, Lecture Notes on Numerical Analysis, Math 5466 (standard reference, not scraped)
Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange Interpolation (standard reference, not scraped)