realPart_unitarySelfAddISMul — Mathlib

English

The real part of the unitarySelfAddISMul a equals a for selfadjoint a with ∥a∥ ≤ 1.

Русский

Действительная часть unitarySelfAddISMul a равна a, если a — самоприводимый и ∥a∥ ≤ 1.

LaTeX

$$$\Re\big( \text{unitarySelfAddISMul } a \big) = a$$$

Lean4

theorem realPart_unitarySelfAddISMul (a : selfAdjoint A) (ha_norm : ‖a‖ ≤ 1) :
    ℜ (unitarySelfAddISMul a ha_norm : A) = a := by
  simp [IsSelfAdjoint.imaginaryPart (x := CFC.sqrt (1 - a ^ 2 : A)) (by cfc_tac)]

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