PA0007

mul_eq_zero

Stable checked-use theorem · independently closed

Zero products have a zero factor: the 23-entry core capstone.

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 n m. n * m = 0 -> n = 0 \/ m = 0

Structural proof guide

Generated structural guide

Zero products have a zero factor: the 23-entry core capstone.

Use the direct prerequisites add_eq_zero_right as previously established PA formulas.

The proof proceeds by structural induction (1), equality transport (1).

Referenced ingredients

PA0004 add_eq_zero_right

Proof neighborhood

Direct dependencies

PA0004 add_eq_zero_right

Direct dependents

PA000I bounded_common_multiple_step PA000S mul_ne_zero PA006S mul_left_cancel_nonzero

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

12 script commands · 9 reading checkpoints · 0 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)
add eq zero right
01Fix variables and assumptionsL1–1

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

  1. L1
    intro n
02Induction on mL2–3

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L2
    induction m
  2. L3
    intro h
03Separate the logical casesL4–4

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

  1. L4
    right
04Calculate and transport equalitiesL5–5

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L5
    refl
05Fix variables and assumptionsL6–6

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

  1. L6
    intro h
06Separate the logical casesL7–7

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

  1. L7
    left
07Use earlier factsL8–10

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

  1. L8
    specialize add_eq_zero_right (n * m)
  2. L9
    specialize add_eq_zero_right n
  3. L10
    apply add_eq_zero_right
08Calculate and transport equalitiesL11–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L11
    rewrite PA6 at h
09Use earlier factsL12–12

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

  1. L12
    exact h

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro n
  2. 0002induction m
  3. 0003intro h
  4. 0004right
  5. 0005refl
  6. 0006intro h
  7. 0007left
  8. 0008specialize add_eq_zero_right (n * m)
  9. 0009specialize add_eq_zero_right n
  10. 0010apply add_eq_zero_right
  11. 0011rewrite PA6 at h
  12. 0012exact h