PA00C9 · theorem

odd_multiplier_parity_iff

Alpha v34 checked-use theorem · independently closed; not Stable

An odd multiplier preserves both parity classes exactly.

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

∀ p. ∀ q. Odd(p) → (Even(p · q) → Even(q)) ∧ (Even(q) → Even(p · q)) ∧ ((Odd(p · q) → Odd(q)) ∧ (Odd(q) → Odd(p · q)))

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0009 Even PD0010 Odd

9 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p q. (exists pod_odd_multiplier. p = 2 * pod_odd_multiplier + 1) -> (((((exists pod_even_product_even. p * q = 2 * pod_even_product_even) -> (exists pod_even_factor_even. q = 2 * pod_even_factor_even)) /\ ((exists pod_even_factor_even. q = 2 * pod_even_factor_even) -> (exists pod_even_product_even. p * q = 2 * pod_even_product_even)))) /\ ((((exists pod_odd_product_odd. p * q = 2 * pod_odd_product_odd + 1) -> (exists pod_odd_factor_odd. q = 2 * pod_odd_factor_odd + 1)) /\ ((exists pod_odd_factor_odd. q = 2 * pod_odd_factor_odd + 1) -> (exists pod_odd_product_odd. p * q = 2 * pod_odd_product_odd + 1)))))

Proof neighborhood

Direct theorem prerequisites

PA00C7 odd_multiplier_even_product_iff PA00C8 odd_multiplier_odd_product_iff

Direct theorem dependents

PA00CK odd_product_division_mod_two

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

12 script commands · 3 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 (2)
odd multiplier even product iffodd multiplier odd product iff
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro hp
02Separate the logical casesL4–4

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

  1. L4
    split
03Use earlier factsL5–12

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

  1. L5
    specialize odd_multiplier_even_product_iff p
  2. L6
    specialize odd_multiplier_even_product_iff q
  3. L7
    apply odd_multiplier_even_product_iff
  4. L8
    exact hp
  5. L9
    specialize odd_multiplier_odd_product_iff p
  6. L10
    specialize odd_multiplier_odd_product_iff q
  7. L11
    apply odd_multiplier_odd_product_iff
  8. L12
    exact hp

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro hp
  4. 0004split
  5. 0005specialize odd_multiplier_even_product_iff p
  6. 0006specialize odd_multiplier_even_product_iff q
  7. 0007apply odd_multiplier_even_product_iff
  8. 0008exact hp
  9. 0009specialize odd_multiplier_odd_product_iff p
  10. 0010specialize odd_multiplier_odd_product_iff q
  11. 0011apply odd_multiplier_odd_product_iff
  12. 0012exact hp