Mathlib v4.30 source proposition Int.Prime.dvd_mul

Identifier: F:ml430-int-prime-dvd-mul-90351ba0
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.Prime.dvd_mul` in the pinned Mathlib v4.30 source.

Formal statement

∀ {m n : ℤ} {p : ℕ}, Nat.Prime p → ↑p ∣ m * n → p ∣ m.natAbs ∨ p ∣ n.natAbs

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Int.prime_dvd_mul

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℤ} {p : ℕ}, Nat.Prime p → ↑p ∣ m * n → p ∣ m.natAbs ∨ p ∣ n.natAbs

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory -- 'prime_dvd_mul' 2>/dev/null | /usr/bin/grep -cE '^theorem[[:space:]]+Int\.prime_dvd_mul[[:space:]]')" -ge 1

Evidence notes

Same route as F:ml430-int-prime-dvd-mul-23b73e69 (Int.prime_dvd_mul'): apply gcd.rs's Int.euclid_lemma at pr := ofNat p to get Or (ofNat p ∣ m) (ofNat p ∣ n), then case-split (Or.rec) and drop each disjunct from Int.dvd to Nat.dvd via the already-proved gcd::declare_nat_abs_dvd_nat_abs_of_dvd (a ∣ b → natAbs a ∣ natAbs b), relying on natAbs (ofNat p) reducing to p by rfl for the final Nat.dvd p (natAbs m)/(natAbs n) shape. int_theorem_inventory's rendered type for Int.prime_dvd_mul matches this fact's formal.statement exactly. New proof, lane int-prime-dvd (2026-09-01): crates/axeyum-lean-kernel/src/int_prelude/prime_dvd_mul_mirrors.rs. No Nat declaration added.

footprint-Int.prime_dvd_mul

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q --release -p axeyum-lean-kernel --example prelude_axiom_inventory -- --require-axiom-free integer

Evidence notes

prelude_axiom_inventory --require-axiom-free integer enumerates the built Int environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: integer: axiom=0, exit 0; the axreal: axiom=30 row is the separate legacy axiomatized-reals package, unrelated to this declaration -- see CLAUDE.md's AxReal/CReal gotcha). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds this declaration's own footprint by []. int_prelude_tests::every_int_declaration_is_checked_and_axiom_free additionally checks this theorem's own Kernel::axiom_footprint directly via derived_lemmas (it is listed there).

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.Prime.dvd_mul` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.Prime.dvd_mul`",
      "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"
    }
  ]
}