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. Exact original first-admission records.

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

Squarefree(n)PowerDivides(p,e,n)PrimeValuationsDivisible(n,k)PrimeExponentPrefixGCD(b,c,l,g)PerfectPowerRootTable(n,g,b,c)PrimeDivisorSupport(n,pb,pc,l)

Checked theorems using this definition

SK0001 · divides_square_of_dividesSK0002 · squarefree_excludes_prime_squareSK0004 · squarefree_squared_divisor_is_oneSK0005 · coprime_squared_pairSK0006 · squarefree_coprime_square_factor_is_oneSK0009 · bounded_prime_square_divisor_searchSK000A · squarefree_or_prime_square_divisorSK000B · squarefree_decomposition_bounded_existsSK000E · squarefree_coprime_square_balanceSK000F · squarefree_decomposition_functionalSK0016 · positive_power_nonzero_baseSK0018 · prime_valuation_divisibility_cofactorSK0019 · prime_valuation_divisible_power_root_boundedSK001B · prime_exponent_common_divisor_dropSK001C · prime_exponent_common_divisor_successorSK001D · prime_exponent_common_divisor_factorSK001F · prime_exponent_prefix_gcd_successorSK0020 · prime_exponent_prefix_gcd_existsSK0023 · prime_valuation_support_exponent_gcd_nonzeroSK0025 · prime_support_common_divisor_implies_all_valuationsSK0026 · prime_support_all_valuations_implies_common_divisorSK0027 · prime_support_exponent_gcd_divisor_criterionSK0028 · prime_support_perfect_power_iff_degree_dividesSK0029 · prime_support_exponent_gcd_roots_availableSK002A · perfect_power_root_table_prefix_appendSK002B · perfect_power_root_table_conditional_entrySK002C · perfect_power_root_table_prefix_existsSK002D · perfect_power_root_table_existsSK002F · perfect_power_profile_existsSK0030 · perfect_power_profile_data_degree_classificationSK0031 · perfect_power_profile_data_root_lookup