inl_mono — Mathlib

English

The projection to the first coordinate in a product is monotone with respect to the product order.

Русский

Проекция на первую координату в произведении мономорона по отношению порядка на произведение.

LaTeX

$$$\text{fst}: M \times N \to M \quad \text{монотонна по продуктному порядку.}$$$

Lean4

theorem inl_mono [LinearOrderedCommGroupWithZero M₀] [GroupWithZero N₀] [Preorder N₀]
    [DecidablePred fun x : M₀ ↦ x = 0] : Monotone (inl M₀ N₀) :=
  by
  refine (WithZero.map'_mono MonoidHom.inl_mono).comp ?_
  intro x y
  obtain rfl | ⟨x, rfl⟩ := GroupWithZero.eq_zero_or_unit x <;>
      obtain rfl | ⟨y, rfl⟩ := GroupWithZero.eq_zero_or_unit y <;>
    · simp [WithZero.withZeroUnitsEquiv]

Похожие утверждения