CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

A finite row reduction from $A$ to $B$ is encoded by $B=E_r\cdots E_1A$

Statement

Suppose a sequence of $r$ elementary row operations transforms $A$ into $B$ , and let $E_1,\ldots,E_r$ be their elementary matrices in execution order. Then $B=E_r\cdots E_1A.$ For $r=0$ the empty product is the identity and $B=A$ .

Facts & Assumptions

Given: A finite sequence of elementary row operations taking $A$ to $B$ .

[L1]

One elementary row operation acts by left multiplication by its elementary matrix (Applying an elementary row operation is left multiplication by its elementary matrix).

[L2]

Compatible matrix multiplication is associative and has identity matrices (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

[L3]

The induction principle proves a property from its zero case and successor step (The principle of mathematical induction).

Proof

technique · induction

1.1

After no operations the intermediate matrix is $A=I_nA$ , the empty-product formula. If after $k$ operations it is $E_k\cdots E_1A$ , then [L1] says the next operation gives $E_{k+1}(E_k\cdots E_1A)$ .

L1L3baseih

2.1

Associativity rewrites the successor expression as $E_{k+1}E_k\cdots E_1A$ ; induction therefore gives the formula after all $r$ operations, including $r=0$ .

step 1.1L2discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.3 (standard reference, not scraped)

A finite row reduction from AAA to BBB is encoded by B=Er⋯E1AB=E_r\cdots E_1AB=Er​⋯E1​A