PD0003

Dvd(d,n)

The natural number d divides n.

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

∃ k. n = d · k

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

Hygienic expanded first-order definition 
exists k. n = d * k

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

Coprime(a,b)HasPrimeSquareDivisor(n)Squarefree(n)DivisorMaskEntry(F,n,d,z)PrimeFactorToggle(p,d,e)DivisorPrimeToggle(n,p,d,e)

Checked theorems using this definition

MC0001 · prime_toggle_square_quotient_dividesMC0002 · prime_toggle_fresh_divisor_productMC0003 · prime_factor_toggle_existsMC0007 · prime_factor_toggle_preserves_divisorMC0008 · divisor_prime_toggle_existsMC000A · divisor_prime_toggle_symmetricMC000B · divisor_prime_toggle_boundedMC000E · divisor_prime_toggle_prefix_permutationMC000F · divisor_prime_toggle_permutation_existsMC0011 · mobius_divisor_mask_actual_valueMC0012 · mobius_divisor_mask_prime_toggle_negatesMC0016 · mobius_divisor_mask_prime_factor_sum_zeroMC0017 · mobius_divisor_sum_nonunit_value_zero