∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] (x : (K →+* ℂ) → ℂ),
  (∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (NumberField.ComplexEmbedding.conjugate φ)) →
    ∀ (c : NumberField.mixedEmbedding.index K),
      ↑(((NumberField.mixedEmbedding.stdBasis K).repr ((NumberField.mixedEmbedding.commMap K) x)) c) =
        (NumberField.mixedEmbedding.matrixToStdBasis K).mulVec (x ∘ ⇑(NumberField.mixedEmbedding.indexEquiv K)) c