BT000M · Bertrand theorem

mul_eq_zero

Stable checked-use theorem · independently kernel verified

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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n m. n * m = 0 -> n = 0 \/ m = 0

Proof neighborhood

Direct theorem prerequisites

BT000L add_eq_zero_right

Direct theorem dependents

BT0021 mul_ne_zero BT0022 mul_left_cancel_nonzero BT0024 two_large_factors_impossible BT004M bounded_common_multiple_step BT006N prime_three

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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