PD0004

Prime(p)

p is nonunit and every factorization of p has a unit factor.

Conservative notation; not a theorem, primitive, or axiom.

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.

Hygienic expanded first-order definition

~(p = 1) /\ forall a b. p = a * b -> a = 1 \/ b = 1

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

none — first-order arithmetic only

Definitions depending on this notation

PrimeThreeModFourDivisor(n,p)AllPrime(b,c,l)

Checked theorems using this definition

TF0004 · beta_all_prime_entry_is_primeTF0005 · beta_admissible_prime_factor_product_is_two_squareTF0006 · positive_number_with_admissible_prime_divisors_is_two_squareTF000A · three_mod_four_good_prime_exclusiveTF0010 · euclid_three_common_multiple_exclusionTF0011 · euclid_three_prime_divisor_exceeds_boundTF0012 · infinitely_many_primes_three_mod_four