Mathlib v4.30 source proposition Nat.Prime.dvd_iff_not_coprime

Identifier: F:ml430-nat-prime-dvd-iff-not-coprime-77854741
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p n : ℕ}, Nat.Prime p → (p ∣ n ↔ ¬p.Coprime n)

Dependencies

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

12 direct dependencies 3 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.prime_dvd_iff_not_coprime

Kind: kernel-term
Status: checked

Supports: for prime p (2 <= p and every divisor of p is 1 or p), p divides n iff gcd p n != 1

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. Forward: `dvd p n` plus `dvd p p` (`dvd_refl`) give `dvd p (gcd p n)` (`dvd_gcd`); if `gcd p n = 1` that transports to `dvd p 1`, forcing `p <= 1` (`le_of_dvd`) against primality's `2 <= p` conjunct -- contradiction via `le_trans`/`le_of_succ_le_succ`/`not_succ_le_zero`, the same numeral-contradiction shape `coprime_of_lt_prime` already uses, so `gcd p n != 1`. Reverse: `gcd p n` divides `p` (`gcd_dvd_left`), so primality's divisor clause forces `gcd p n = 1 or gcd p n = p`; the hypothesis rules out `= 1`, leaving `gcd p n = p`, and `dvd p n` follows from `gcd_dvd_right` transported along that equation. `Nat.Prime` has no separate name in this kernel; it is spelled inline as the same `2 <= p and every divisor is 1 or p` clause `coprime_of_lt_prime` already uses. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` requires the admitted declaration to be printed.

footprint-Nat.prime_dvd_iff_not_coprime

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. 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": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane coprime-backlog",
  "source": "statement-only extraction of `Nat.Prime.dvd_iff_not_coprime` from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.prime_dvd_iff_not_coprime` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Prime.dvd_iff_not_coprime`",
      "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, and this project's proof is independently constructed"
    }
  ]
}