PA006X · theorem

distinct_primes_mutually_nondivisible

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

Distinct primes are mutually nondivisible.

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) ∧ ¬Dvd(q,p)

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

4 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)) /\ (~(exists frm_factor_dpn_q_not_p. p = q * frm_factor_dpn_q_not_p)))

Proof neighborhood

Direct theorem prerequisites

PA006V distinct_primes_left_not_divide_right PA006W distinct_primes_right_not_divide_left

Direct theorem dependents

PA00FG distinct_odd_primes_gauss_eisenstein_data_exists

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

22 script commands · 7 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)
distinct primes left not divide rightdistinct primes right not divide left
01Fix variables and assumptionsL1–5

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
02Separate the logical casesL6–6

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

  1. L6
    split
03Use earlier factsL7–8

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

  1. L7
    specialize distinct_primes_left_not_divide_right p
  2. L8
    specialize distinct_primes_left_not_divide_right q
04Fix variables and assumptionsL9–9

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

  1. L9
    intro hdiv
05Use earlier factsL10–16

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

  1. L10
    apply distinct_primes_left_not_divide_right
  2. L11
    exact hp
  3. L12
    exact hq
  4. L13
    exact hpq
  5. L14
    exact hdiv
  6. L15
    specialize distinct_primes_right_not_divide_left p
  7. L16
    specialize distinct_primes_right_not_divide_left q
06Fix variables and assumptionsL17–17

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

  1. L17
    intro hdiv
07Use earlier factsL18–22

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

  1. L18
    apply distinct_primes_right_not_divide_left
  2. L19
    exact hp
  3. L20
    exact hq
  4. L21
    exact hpq
  5. L22
    exact hdiv

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro hp
  4. 0004intro hq
  5. 0005intro hpq
  6. 0006split
  7. 0007specialize distinct_primes_left_not_divide_right p
  8. 0008specialize distinct_primes_left_not_divide_right q
  9. 0009intro hdiv
  10. 0010apply distinct_primes_left_not_divide_right
  11. 0011exact hp
  12. 0012exact hq
  13. 0013exact hpq
  14. 0014exact hdiv
  15. 0015specialize distinct_primes_right_not_divide_left p
  16. 0016specialize distinct_primes_right_not_divide_left q
  17. 0017intro hdiv
  18. 0018apply distinct_primes_right_not_divide_left
  19. 0019exact hp
  20. 0020exact hq
  21. 0021exact hpq
  22. 0022exact hdiv