cancel_left — Mathlib

English

If g: SupBotHom β γ is injective, then g ∘ f1 = g ∘ f2 iff f1 = f2 for f1, f2: SupBotHom α β.

Русский

Если g: SupBotHom(β, γ) инъективен, то g ∘ f1 = g ∘ f2 эквивалентно f1 = f2 для f1, f2: SupBotHom(α, β).

LaTeX

$$$\\forall f_1,f_2:\\mathrm{SupBotHom}(\\alpha,\\beta),\\ \\text{Injective}(g)\\implies (g\\circ f_1 = g\\circ f_2 \\iff f_1=f_2)$$$

Lean4

@[simp]
theorem cancel_left {g : SupBotHom β γ} {f₁ f₂ : SupBotHom α β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
  ⟨fun h => SupBotHom.ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩

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