BT003M

prime_divisor_eq_one_or_self

Stable · empty-context checked

Every divisor of a prime is one or the prime itself.

Exact expanded PA statement

forall p g. (~(p = 1) /\ forall c d. p = c * d -> c = 1 \/ d = 1) -> (exists x. p = g * x) -> g = 1 \/ p = g

Structural proof guide

Every divisor of a prime is one or the prime itself.

Direct prerequisites: mul_one. The authored body proceeds by case analysis (3), intermediate claims (1), equality transport (1).

Proof neighborhood

Direct dependencies

BT000A mul_one

Direct dependents

BT003N euclid_prime_dvd_product BT008Q prime_coprime_or_divides BT008S distinct_primes_coprime

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

  1. 0001intro p
  2. 0002intro g
  3. 0003intro hp
  4. 0004intro hdiv
  5. 0005cases hp
  6. 0006cases hdiv
  7. 0007specialize hp_right g
  8. 0008specialize hp_right x
  9. 0009have hfactor : g = 1 \/ x = 1
  10. 0010apply hp_right
  11. 0011exact hdiv_witness
  12. 0012cases hfactor
  13. 0013left
  14. 0014exact hfactor_left
  15. 0015right
  16. 0016trans g * x
  17. 0017exact hdiv_witness
  18. 0018rewrite hfactor_right
  19. 0019apply mul_one