Mathlib v4.30 source proposition Nat.Prime.dvd_lcm

Identifier: F:ml430-nat-prime-dvd-lcm-237d267c
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p a b : ℕ}, Nat.Prime p → (p ∣ a.lcm b ↔ p ∣ a ∨ p ∣ b)

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

10 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.prime_dvd_lcm_iff

Kind: kernel-term
Status: checked

Supports: ∀ {p a b : ℕ}, Nat.Prime p → (p ∣ a.lcm b ↔ p ∣ a ∨ p ∣ b)

Checker command

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

Evidence notes

`declare_prime_dvd_lcm_iff` (`nat_prelude/prime_dvd_factorial_lcm.rs`) admits `Nat.prime_dvd_lcm_iff` through the trusted `Kernel::add_declaration` gate. Pure algebra, no induction: `mp` uses `dvd_mul_left_of_dvd` to extend `p | lcm a b` to `p | (gcd a b * lcm a b)`, `gcd_mul_lcm` (`gcd a b * lcm a b = a * b`, unconditional) identifies that product with `a*b`, and `euclid_lemma` splits `p | a*b` into `p | a \/ p | b`. `mpr` is `dvd_trans` through `dvd_lcm_left`/`dvd_lcm_right`, case-split over the `Or` -- primality is unused on this side, only carried because the statement's hypothesis wraps the whole `Iff`.

footprint-Nat.prime_dvd_lcm_iff

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