one_apply_apply_eq — Mathlib

English

The composition of the constant-1 hom with any MonoidWithZeroHom equals the constant-1 hom on the domain.

Русский

Композиция константной единицы с MonoidWithZeroHom равна константной единице на области.

LaTeX

$$$(1 : N \to*₀ G) \circ f = (1 : M \to*₀ G)$$$

Lean4

/-- The trivial group-with-zero hom is absorbing for composition. -/
@[simp]
theorem one_apply_apply_eq {M₀ N₀ G₀ : Type*} [GroupWithZero M₀] [MulZeroOneClass N₀] [Nontrivial N₀]
    [NoZeroDivisors N₀] [MulZeroOneClass G₀] [DecidablePred fun x : M₀ ↦ x = 0] [DecidablePred fun x : N₀ ↦ x = 0]
    (f : M₀ →*₀ N₀) (x : M₀) : (1 : N₀ →*₀ G₀) (f x) = (1 : M₀ →*₀ G₀) x :=
  by
  rcases eq_or_ne x 0 with rfl | hx
  · simp
  · rw [one_apply_of_ne_zero hx, one_apply_of_ne_zero]
    rwa [map_ne_zero f]

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