hom_ext — Mathlib

English

If g1,g2 : WidePushout → X have equal compositions with all injections and with head, then g1 = g2.

Русский

Если g1 и g2 : WidePushout → X совпадают после композиции с каждым ι arrows j и head arrows, тогда g1 = g2.

LaTeX

$$$(\\forall j:\\, J,\\; ι arrows j \\circ g1 = ι arrows j \\circ g2) \\to (head arrows \\circ g1 = head arrows \\circ g2) \\Rightarrow g1 = g2$.$$

Lean4

@[ext 1100]
theorem hom_ext (g1 g2 : widePushout _ _ arrows ⟶ X) :
    (∀ j : J, ι arrows j ≫ g1 = ι arrows j ≫ g2) → head arrows ≫ g1 = head arrows ≫ g2 → g1 = g2 :=
  by
  intro h1 h2
  apply colimit.hom_ext
  rintro (_ | _)
  · apply h2
  · apply h1

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