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.

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

none

Checked theorems using this definition

CG0001 · mod_eq_cancel_gcd_cofactorCG0002 · linear_congruence_solution_class_iff_reduced_modulusCG0003 · linear_congruence_reduced_representative_existsCG0007 · linear_congruence_bounded_solutions_parametrizedCG0008 · linear_congruence_exact_bounded_enumeration_exists