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

Squarefree(n)PrimeValuationsDivisible(n,k)PrimeExponentEntries(n,pb,pc,eb,ec,vb,vc,l)PrimeDivisorSupport(n,pb,pc,l)AllPrime(b,c,l)

Checked theorems using this definition

SK0002 · squarefree_excludes_prime_squareSK0003 · prime_square_ne_oneSK0004 · squarefree_squared_divisor_is_oneSK0009 · bounded_prime_square_divisor_searchSK000A · squarefree_or_prime_square_divisorSK000B · squarefree_decomposition_bounded_existsSK0018 · prime_valuation_divisibility_cofactorSK0019 · prime_valuation_divisible_power_root_boundedSK0023 · prime_valuation_support_exponent_gcd_nonzeroSK0024 · prime_exponent_entry_has_prime_valuationSK0025 · prime_support_common_divisor_implies_all_valuationsSK0026 · prime_support_all_valuations_implies_common_divisor