isUnit_iff_bijective — Mathlib

English

Intermediate construction: an isomorphism between the coproduct E × G.subtypeL and F when range(f) is complementary to G.

Русский

Промежуточная конструкция: изоморфизм между копро(d) E × G и F при условии, что range(f) комплементарно G.

LaTeX

$$$(E \\times G) \\cong F$ при IsCompl(range f, G)$$

Lean4

theorem _root_.ContinuousLinearMap.isUnit_iff_bijective {f : E →L[𝕜] E} : IsUnit f ↔ Bijective f :=
  by
  constructor
  · rintro ⟨f, rfl⟩
    exact ofUnit f |>.bijective
  ·
    refine fun h ↦ ⟨toUnit <| .ofBijective f ?_ ?_, rfl⟩ <;>
      simp only [LinearMap.range_eq_top, LinearMapClass.ker_eq_bot, h.1, h.2]

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