starMap_injective — Mathlib

English

Simplified equivalence of self-adjointness under inr embedding.

Русский

Упрощённая эквивалентность самосопряженности через вложение inr.

LaTeX

$$$IsSelfAdjoint( inr(a) ) \Leftrightarrow IsSelfAdjoint(a).$$$

Lean4

/-- If `φ : A →⋆ₙₐ[R] B` is injective, the lift `starMap φ : Unitization R A →⋆ₐ[R] Unitization R B`
is also injective. -/
theorem starMap_injective {φ : A →⋆ₙₐ[R] B} (hφ : Function.Injective φ) : Function.Injective (starMap φ) :=
  by
  intro x y h
  ext
  · simpa using congr(fst $(h))
  · exact hφ <| by simpa [algebraMap_eq_inl] using congr(snd $(h))

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