ofBijective_symm_comp — Mathlib

English

If f is a bijective nonunital ring hom, then composing its RingEquiv.ofBijective symmetry with f yields the identity, i.e., the inverse composition equals id.

Русский

Если f — биективный нена единичный кольцевой гомоморфизм, то композиция его RingEquiv.ofBijective симм с f даёт тождественное отображение.

LaTeX

$$$f \\circ (\\mathrm{RingEquiv.ofBijective}(f))^{ -1} = \\mathrm{id}$$$

Lean4

@[simp]
theorem ofBijective_symm_comp (f : R →ₙ+* S) (hf : Function.Bijective f) :
    ((RingEquiv.ofBijective f hf).symm : _ →ₙ+* _).comp f = NonUnitalRingHom.id R :=
  by
  ext
  exact (RingEquiv.ofBijective f hf).injective <| RingEquiv.apply_symm_apply ..

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