cancel_right — Mathlib

English

If f is surjective, then composing on the right with f is injective on the right: g1 ∘ f = g2 ∘ f implies g1 = g2.

Русский

Если f сюрьективно, то right-cancellation holds: g1 ∘ f = g2 ∘ f ⇒ g1 = g2.

LaTeX

$$$\\text{Surjective}(f)\\;\\Rightarrow\\; g_1\\circ f = g_2\\circ f \\iff g_1 = g_2.$$$

Lean4

@[simp]
theorem cancel_right {g₁ g₂ : TopHom β γ} {f : TopHom α β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
  ⟨fun h => TopHom.ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, congr_arg (fun g => comp g f)⟩

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