Mathlib v4.30 source proposition Nat.Prime.mod_two_eq_one_iff_ne_two

Identifier: F:ml430-nat-prime-mod-two-eq-one-iff-ne-two-25c35e73
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p : ℕ}, Nat.Prime p → (p % 2 = 1 ↔ p ≠ 2)

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.prime_mod_two_eq_one_iff_ne_two

Kind: kernel-term
Status: checked

Supports: ∀ {p : ℕ}, Nat.Prime p → (p % 2 = 1 ↔ p ≠ 2)

Checker command

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

Evidence notes

`mp` needs no primality: assuming `p=2`, transport `mod p 2=1` along it to `mod 2 2=1`, contradicting `mod 2 2=0` (`even_iff_mod_two_eq_zero` applied to a hand-built `Even 2` witness, `Exists.intro 1 (Eq.refl 2)`) via `succ_ne_zero`. `mpr` is `prime_eq_two_or_mod_two_eq_one`'s case split, with the `Even` branch now contradicting the hypothesis instead of closing the goal.

footprint-Nat.prime_mod_two_eq_one_iff_ne_two

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 `Nat.prime_mod_two_eq_one_iff_ne_two`'s footprint by [].

Provenance

{
  "date": "2026-08-30",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane prime-dvd-mirrors",
  "source": "statement-only extraction of `Nat.Prime.mod_two_eq_one_iff_ne_two` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Prime.mod_two_eq_one_iff_ne_two`",
      "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"
    }
  ]
}