sumSet — Mathlib

English

For disjoint s and t, the sumSet h is the embedding from s ⊕ t into α given by Sum.elim.

Русский

Для несжимаемых s и t суммаSet h — вложение из s ⊕ t в α, заданное Sum.elim.

LaTeX

$$$\\text{sumSet } h : s \\oplus t \\hookrightarrow \\alpha$$$

Lean4

/-- For disjoint `s t : Set α`, the natural injection from `↑s ⊕ ↑t` to `α`. -/
@[simps]
def sumSet (h : Disjoint s t) : s ⊕ t ↪ α where
  toFun := Sum.elim (↑) (↑)
  inj' := by
    rintro (⟨a, ha⟩ | ⟨a, ha⟩) (⟨b, hb⟩ | ⟨b, hb⟩)
    · simp
    · simpa using h.ne_of_mem ha hb
    · simpa using h.symm.ne_of_mem ha hb
    simp

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