comap_injective — Mathlib

English

If f ∘ g = id, then ker f = range(id − g ∘ f).

Русский

Если f ∘ g = id, то ker f = range(id − g ∘ f).

LaTeX

$$$f \circ g = \mathrm{id} \Rightarrow \ker f = \operatorname{range}(\mathrm{id} - g \circ f)$$$

Lean4

theorem comap_injective {f : F} (hf : range f = ⊤) : Injective (comap f) := fun _ _ h =>
  le_antisymm ((comap_le_comap_iff hf).1 (le_of_eq h))
    ((comap_le_comap_iff hf).1 (ge_of_eq h))
      -- TODO (?): generalize to semilinear maps with `f ∘ₗ g` bijective.

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