BT0021

mul_ne_zero

Stable checked-use theorem · independently kernel verified

A product of two nonzero naturals is nonzero.

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 b. ~(a = 0) -> ~(b = 0) -> ~(a * b = 0)

Structural proof guide

A product of two nonzero naturals is nonzero.

Direct prerequisites: mul_eq_zero. The authored body proceeds by case analysis (1), intermediate claims (1).

Proof neighborhood

Direct dependencies

BT000M mul_eq_zero

Direct dependents

BT0022 mul_left_cancel_nonzero BT0059 beta_exclusive_accumulated_product_step BT00QR power_valuation_mul_lower BT00RK factorial_nonzero BT00XM central_binom_factorial_valuation_balance

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

15 script commands · 5 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 (1)
mul eq zero
01Fix variables and assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro ha
  4. L4
    intro hb
  5. L5
    intro hab
02Use earlier factsL6–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L6
    specialize mul_eq_zero a
  2. L7
    specialize mul_eq_zero b
03Establish hzL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul eq zero.

  1. L8
    have hz : a = 0 \/ b = 0
  2. L9
    apply mul_eq_zero
  3. L10
    exact hab
04Separate the logical casesL11–11

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases hz
05Use earlier factsL12–15

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L12
    apply ha
  2. L13
    exact hz_left
  3. L14
    apply hb
  4. L15
    exact hz_right

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro ha
  4. 0004intro hb
  5. 0005intro hab
  6. 0006specialize mul_eq_zero a
  7. 0007specialize mul_eq_zero b
  8. 0008have hz : a = 0 \/ b = 0
  9. 0009apply mul_eq_zero
  10. 0010exact hab
  11. 0011cases hz
  12. 0012apply ha
  13. 0013exact hz_left
  14. 0014apply hb
  15. 0015exact hz_right