Mathlib v4.30 source proposition Nat.Prime.dvd_mul_of_dvd_ne

Identifier: F:ml430-nat-prime-dvd-mul-of-dvd-ne-6c253439
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.Prime.dvd_mul_of_dvd_ne` in the pinned Mathlib v4.30 source.

Formal statement

∀ {p1 p2 n : ℕ}, p1 ≠ p2 → Nat.Prime p1 → Nat.Prime p2 → p1 ∣ n → p2 ∣ n → p1 * p2 ∣ n

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.prime_dvd_mul_of_dvd_ne

Kind: kernel-term
Status: checked

Supports: ∀ {p1 p2 n : ℕ}, p1 ≠ p2 → Nat.Prime p1 → Nat.Prime p2 → p1 ∣ n → p2 ∣ n → p1 * p2 ∣ n

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- prime_dvd_mul_of_dvd_ne 2>/dev/null | grep -Ec '^Nat\.prime_dvd_mul_of_dvd_ne[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.prime_dvd_mul_of_dvd_ne` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Primality is spelled inline (`2 ≤ p ∧ ∀ d, d ∣ p → d = 1 ∨ d = p`) rather than through a named `Nat.Prime` predicate. `coprime_primes`'s `mpr` direction turns the `p1 ≠ p2` hypothesis (with both primality hypotheses) into `Coprime p1 p2`, then `coprime_mul_dvd` (`crt.rs`) combines the two divisibility hypotheses into `dvd (mul p1 p2) n`. Declared after `declare_crt` in the build pipeline for that reason.

footprint-Nat.prime_dvd_mul_of_dvd_ne

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.Prime.dvd_mul_of_dvd_ne` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Prime.dvd_mul_of_dvd_ne`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}