BT002B

multiple_mul_left

Stable checked-use theorem · independently kernel verified

A left multiple of a multiple remains a multiple.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded PA statement

forall a n m. (exists q. n = a * q) -> exists s. m * n = a * s

Structural proof guide

A left multiple of a multiple remains a multiple.

Direct prerequisites: mul_comm, multiple_mul_right. The authored body proceeds by equality transport (1).

Proof neighborhood

Direct dependencies

BT0006 mul_comm BT002A multiple_mul_right

Direct dependents

BT0101 prime_contribution_selected_successor_divides BT0102 prime_contribution_cofactor_prime_contradiction

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

9 script commands · 2 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (2)
mul commmultiple mul right
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro n
  3. L3
    intro m
  4. L4
    intro hn
02Reduce to hswapL5–9

First show that this intermediate claim suffices; its proof is a separate obligation.

  1. L5
    suffices hswap : m * n = n * m
  2. L6
    rewrite hswap
  3. L7
    apply multiple_mul_right
  4. L8
    exact hn
  5. L9
    apply mul_comm

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002intro n
  3. 0003intro m
  4. 0004intro hn
  5. 0005suffices hswap : m * n = n * m
  6. 0006rewrite hswap
  7. 0007apply multiple_mul_right
  8. 0008exact hn
  9. 0009apply mul_comm