mul — Mathlib

English

The negation of monic 0 is equivalent to 0 ≠ 1.

Русский

Отрицание моничности нулевого полинома эквивалентно 0 ≠ 1.

LaTeX

$$$\neg\operatorname{Monic}(0) \iff 0 \neq 1.$$$

Lean4

theorem mul (hp : Monic p) (hq : Monic q) : Monic (p * q) :=
  letI := Classical.decEq R
  if h0 : (0 : R) = 1 then
    haveI := subsingleton_of_zero_eq_one h0
    Subsingleton.elim _ _
  else by
    have : p.leadingCoeff * q.leadingCoeff ≠ 0 := by simp [Monic.def.1 hp, Monic.def.1 hq, Ne.symm h0]
    rw [Monic.def, leadingCoeff_mul' this, Monic.def.1 hp, Monic.def.1 hq, one_mul]

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