Mathlib v4.30 source proposition Nat.Coprime.mul_add_mul_ne_mul

Identifier: F:ml430-nat-coprime-mul-add-mul-ne-mul-51b56f70
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.Coprime.mul_add_mul_ne_mul` in the pinned Mathlib v4.30 source.

Formal statement

∀ {m n a b : ℕ}, m.Coprime n → a ≠ 0 → b ≠ 0 → a * m + b * n ≠ m * n

Dependencies

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21 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.coprime_mul_add_mul_ne_mul

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Nat.Coprime.mul_add_mul_ne_mul` in the pinned Mathlib v4.30 source.

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::

Evidence notes

`declare_coprime_mul_add_mul_ne_mul` (`crates/axeyum-lean-kernel/src/nat_prelude/coprime_mul_add_mul_ne_mul.rs`): a `cases_zero_succ` split on `m` (then on `n`, inside the `m>=1` branch) for the `m=0`/`n=0` degenerate cases, and Gauss's lemma run in both directions (`m|b`, `n|a` via `gauss_lemma`/`coprime_symmetric`, then `le_of_dvd`/`mul_le_mul_left`/`one_le_mul`/`lt_irrefl` for the arithmetic contradiction) in the `m,n>=1` case. Registered here as `Nat.coprime_mul_add_mul_ne_mul` (lowercase-flattened, matching this codebase's existing convention for `Nat.Coprime.*` methods such as `coprime_symmetric`).

footprint-Nat.coprime_mul_add_mul_ne_mul

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- this theorem's trusted closure is empty

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.coprime_mul_add_mul_ne_mul 2>/dev/null | grep -Ec 'nat[[:space:]]Nat\.coprime_mul_add_mul_ne_mul[[:space:]]0[[:space:]]*$')" -ge 1

Evidence notes

The matched row's trailing footprint column is EMPTY, so a nonzero footprint or an absent declaration both give a count of 0 and fail. The query name must be `Nat.coprime_mul_add_mul_ne_mul` (lowercase `c`), NOT `Nat.Coprime.mul_add_mul_ne_mul` -- the tool matches by exact string, and the latter surfaces zero rows.

Provenance

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