pointsPi_injective — Mathlib

English

Same as 37919: The map pointsPi is injective for quasi-separated X.

Русский

Повторение: pointsPi инъективно при квазиразделимой X.

LaTeX

$$Injectivity of pointsPi: If X is quasi-separated, then pointsPi R X is injective.$$

Lean4

theorem pointsPi_injective [QuasiSeparatedSpace X] : Function.Injective (pointsPi R X) :=
  by
  rintro f g e
  have :=
    isIso_of_comp_eq_sigmaSpec R (V := equalizer f g)
      (equalizer.lift (sigmaSpec R) (by ext1 i; simpa using congr_fun e i)) (equalizer.ι f g) (by simp)
  rw [← cancel_epi (equalizer.ι f g), equalizer.condition]

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