PD0006

IsGCD(g,a,b)

g is a common divisor divisible by every common divisor.

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

Dvd(g,a) ∧ Dvd(g,b) ∧ (∀ x. Dvd(x,a) → Dvd(x,b) → Dvd(x,g))

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

Hygienic expanded first-order definition 
((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. g = d * w

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

Direct definition dependencies

Dvd(d,n)

Definitions depending on this notation

CRTPrefixGcdCongruences(b,c,l,m,u,v)CRTPairwiseCompatiblePrefix(r,s,b,c,l)

Checked theorems using this definition

FC0001 · crt_gcd_zero_right_valueFC0002 · crt_gcd_nonzero_leftFC0003 · crt_gcd_nonzero_rightFC0004 · crt_gcd_coprime_cofactorsFC0005 · crt_gcd_lcm_distributes_scaled_coprimeFC0006 · crt_gcd_lcm_distributes_nonzeroFC0007 · crt_gcd_lcm_distributes_zero_leftFC0008 · crt_gcd_lcm_distributes_zero_comparisonFC0009 · crt_gcd_lcm_distributesFC000B · crt_prefix_gcd_congruences_lcmFC000D · crt_pairwise_compatible_prefix_implies_merge_compatibleFC0015 · crt_pairwise_compatible_prefix_merge_iff