range_cfc_eq_range_cfcHom — Mathlib

English

The star operation commutes with cfc: cfc (star ·) a equals star (cfc f a).

Русский

Операция сопряжения commuting с cfc: cfc (star ·) a = star (cfc f a).

LaTeX

$$cfc (\text{star} \cdot) a = \text{star} (cfc f a)$$

Lean4

theorem range_cfc_eq_range_cfcHom [StarModule R A] {a : A} (ha : p a) :
    Set.range (cfc (R := R) · a) = (cfcHom ha (R := R)).range :=
  by
  ext
  constructor
  all_goals rintro ⟨f, rfl⟩
  · exact cfc_cases _ a f (zero_mem _) fun hf ha ↦ ⟨_, rfl⟩
  · exact ⟨Subtype.val.extend f 0, cfcHom_eq_cfc_extend _ ha _ |>.symm⟩

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