PA006V · theorem

distinct_primes_left_not_divide_right

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

A prime cannot divide a distinct prime.

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. Prime(p) → Prime(q) → ¬p = q → ¬Dvd(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

PD0003 Dvd PD0004 Prime

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p q. ((~(p = 1) /\ forall frm_prime_left_dpn_prime_p frm_prime_right_dpn_prime_p. p = frm_prime_left_dpn_prime_p * frm_prime_right_dpn_prime_p -> frm_prime_left_dpn_prime_p = 1 \/ frm_prime_right_dpn_prime_p = 1)) -> ((~(q = 1) /\ forall frm_prime_left_dpn_prime_q frm_prime_right_dpn_prime_q. q = frm_prime_left_dpn_prime_q * frm_prime_right_dpn_prime_q -> frm_prime_left_dpn_prime_q = 1 \/ frm_prime_right_dpn_prime_q = 1)) -> ~(p = q) -> (~(exists frm_factor_dpn_p_not_q. q = p * frm_factor_dpn_p_not_q))

Proof neighborhood

Direct theorem prerequisites

PA0003 prime_divisor_eq_one_or_self

Direct theorem dependents

PA006W distinct_primes_right_not_divide_left PA006X distinct_primes_mutually_nondivisible

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

19 script commands · 6 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.

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)
prime divisor eq one or self
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro hp
  4. L4
    intro hq
  5. L5
    intro hpq
  6. L6
    intro hdiv
02Establish hfactorL7–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime divisor eq one or self.

  1. L7
    have hfactor : p = 1 \/ q = p
  2. L8
    specialize prime_divisor_eq_one_or_self q
  3. L9
    specialize prime_divisor_eq_one_or_self p
  4. L10
    apply prime_divisor_eq_one_or_self
  5. L11
    exact hq
  6. L12
    exact hdiv
03Separate the logical casesL13–14

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

  1. L13
    cases hfactor
  2. L14
    cases hp
04Use earlier factsL15–17

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

  1. L15
    apply hp_left
  2. L16
    exact hfactor_left
  3. L17
    apply hpq
05Calculate and transport equalitiesL18–18

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

  1. L18
    symm
06Use earlier factsL19–19

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

  1. L19
    exact hfactor_right

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro hp
  4. 0004intro hq
  5. 0005intro hpq
  6. 0006intro hdiv
  7. 0007have hfactor : p = 1 \/ q = p
  8. 0008specialize prime_divisor_eq_one_or_self q
  9. 0009specialize prime_divisor_eq_one_or_self p
  10. 0010apply prime_divisor_eq_one_or_self
  11. 0011exact hq
  12. 0012exact hdiv
  13. 0013cases hfactor
  14. 0014cases hp
  15. 0015apply hp_left
  16. 0016exact hfactor_left
  17. 0017apply hpq
  18. 0018symm
  19. 0019exact hfactor_right