rightInverse_of_card_le — Mathlib

English

If f: α → β has a left inverse g and card α ≤ card β, then g is a right inverse of f.

Русский

Если существует левый обратный g к f и card α ≤ card β, то g является правой обратной к f.

LaTeX

$$$\\text{LeftInverse } f\ household g \\land \\operatorname{card} \\alpha \\le \\operatorname{card} \\beta \\Rightarrow \\text{RightInverse } f g$$$

Lean4

theorem _root_.Function.LeftInverse.rightInverse_of_card_le {f : α → β} {g : β → α} (hfg : LeftInverse f g)
    (hcard : card α ≤ card β) : RightInverse f g :=
  have hsurj : Surjective f := surjective_iff_hasRightInverse.2 ⟨g, hfg⟩
  rightInverse_of_injective_of_leftInverse
    ((bijective_iff_surjective_and_card _).2 ⟨hsurj, le_antisymm hcard (card_le_of_surjective f hsurj)⟩).1 hfg

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