map_surjective — Mathlib

English

If f and g are surjective, then Prod.map f g is surjective.

Русский

Если f и g сюръективны, то Prod.map f g сюръективен.

LaTeX

$$Surjective (Prod.map f g)$$

Lean4

@[simp]
theorem map_surjective [Nonempty γ] [Nonempty δ] {f : α → γ} {g : β → δ} :
    Surjective (map f g) ↔ Surjective f ∧ Surjective g :=
  ⟨fun h =>
    ⟨fun c => by
      inhabit δ
      obtain ⟨⟨a, b⟩, h⟩ := h (c, default)
      exact ⟨a, congr_arg Prod.fst h⟩, fun d => by
      inhabit γ
      obtain ⟨⟨a, b⟩, h⟩ := h (default, d)
      exact ⟨b, congr_arg Prod.snd h⟩⟩,
    fun h => h.1.prodMap h.2⟩

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