Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Multiplication by zero: 0⋅a=0

Statement

In any field F (Field), for every a∈F we have 0⋅a=0.

Statement references

Field

Facts & Assumptions

Given: A field F and an element a∈F.

[L1]

Field axioms (Field): 0 is the additive identity, so 0+0=0 and x+0=x for all x; multiplication distributes over addition, (y+z)⋅x=y⋅x+z⋅x.

[L2]

Field axioms (Field): every element has an additive inverse, so addition admits cancellation: if x+z=y+z then x=y.

Proof

technique · direct
1.1L1

Since 0+0=0, we have 0⋅a=(0+0)⋅a.

1.2L1

By distributivity, (0+0)⋅a=0⋅a+0⋅a.

1.3L1

Since 0 is the additive identity, 0⋅a=0⋅a+0.

2.1step 1.1step 1.2

Combining the two expressions for (0+0)⋅a gives 0⋅a=0⋅a+0⋅a.

3.1step 1.3step 2.1

From steps 1.3 and 2.1, 0⋅a+0=0⋅a+0⋅a.

4.1step 3.1L2∎

Cancelling 0⋅a from both sides yields 0=0⋅a, that is 0⋅a=0.

Depends on

Dependency tree · 1 level

Sources