`BookProof.Complexification.Cx.inner_im` (x y : Cx W) : RCLike.im (inner ℂ x y) = inner ℝ x.re y.im - inner ℝ x.im y.re

Proved

BookProof.Complexification.Cx.inner_im

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from Complexification.

BookProof.Complexification.Cx.inner_im (x y : Cx W) : RCLike.im (inner ℂ x y) = inner ℝ x.re y.im - inner ℝ x.im y.re

Formalization note: Lean 4 identifier BookProof.Complexification.Cx.inner_im.

Preamble

-- Generated from Complexification.lean — theorem BookProof.Complexification.Cx.inner_im
import Mathlib
import Definitions.Def_Complexification
open BookProof.Complexification
open BookProof.Complexification


open scoped RealInnerProductSpace
open RCLike


set_option linter.unusedSectionVars false

variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W]

Formal statement

theorem BookProof.Complexification.Cx.inner_im (x y : Cx W) :
    RCLike.im (inner ℂ x y) = inner ℝ x.re y.im - inner ℝ x.im y.re := by sorry

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