PA006W · theorem

distinct_primes_right_not_divide_left

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

The reverse orientation is nondivisible as well.

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(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

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_q_not_p. p = q * frm_factor_dpn_q_not_p))

Proof neighborhood

Direct theorem prerequisites

PA006V distinct_primes_left_not_divide_right

Direct theorem dependents

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

18 script commands · 3 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)
distinct primes left not divide right
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
02Establish hqpL6–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hpq.

  1. L6
    have hqp : ~(q = p)
  2. L7
    intro h
  3. L8
    apply hpq
  4. L9
    symm
  5. L10
    exact h
  6. L11
    specialize distinct_primes_left_not_divide_right q
  7. L12
    specialize distinct_primes_left_not_divide_right p
  8. L13
    intro hdiv
  9. L14
    apply distinct_primes_left_not_divide_right
  10. L15
    exact hq
03Use earlier factsL16–18

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

  1. L16
    exact hp
  2. L17
    exact hqp
  3. L18
    exact hdiv

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro hp
  4. 0004intro hq
  5. 0005intro hpq
  6. 0006have hqp : ~(q = p)
  7. 0007intro h
  8. 0008apply hpq
  9. 0009symm
  10. 0010exact h
  11. 0011specialize distinct_primes_left_not_divide_right q
  12. 0012specialize distinct_primes_left_not_divide_right p
  13. 0013intro hdiv
  14. 0014apply distinct_primes_left_not_divide_right
  15. 0015exact hq
  16. 0016exact hp
  17. 0017exact hqp
  18. 0018exact hdiv