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.

Definition in prerequisite notation

¬p = 1 ∧ (∀ x. ∀ y. p = x · y → x = 1 ∨ y = 1)

Only definitions earlier in this acyclic notation graph are used here.

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

EulerPrimePowerFactor(p,e,c)PrimeExponentEntries(n,pb,pc,eb,ec,vb,vc,l)PrimeDivisorSupport(n,pb,pc,l)AllPrime(b,c,l)

Checked theorems using this definition

TP0029 · totient_prime_coprime_iff_nondivisorTP002C · totient_prime_multiple_is_not_unitTP002E · totient_prime_block_positive_offset_unitsTP0033 · totient_new_prime_block_count_balanceTP0034 · totient_new_prime_multiple_blocks_balanceTP0035 · totient_unit_count_new_prime_balanceTP0036 · totient_unit_count_new_prime_factorTP0037 · totient_repeated_prime_factorTP0038 · totient_new_prime_factorTP0039 · totient_prime_valueTP003A · totient_prime_power_successor_valueTP003B · totient_prime_power_valueTP003D · totient_coprime_multiplication_prime_stepTP003E · totient_coprime_multiplicative_from_prime_listTP0040 · totient_euler_factor_existsTP0043 · totient_prime_entries_decoded_powerTP0044 · totient_prime_entries_selected_powerTP0045 · totient_prime_support_powers_pairwise_coprimeTP004A · totient_euler_factor_prefix_exists