skewAdjointPart_comp_subtype_skewAdjoint

English

The composition of the self-adjoint part with the skew-adjoint subtype is the identity.

Русский

Состав самосопряжённой части с подтипом космодружённых даёт тождество.

LaTeX

$$$(\text{selfAdjointPart})(\text{skewAdjoint.submodule } R A).subtype = \mathrm{id}$$$

Lean4

@[simp]
theorem skewAdjointPart_comp_subtype_skewAdjoint : (skewAdjointPart R).comp (skewAdjoint.submodule R A).subtype = .id :=
  LinearMap.ext fun ⟨x, (hx : _ = _)⟩ ↦
    Subtype.eq <| by
      simp only [LinearMap.comp_apply, Submodule.subtype_apply, skewAdjointPart_apply_coe, hx, sub_neg_eq_add, smul_add,
        invOf_two_smul_add_invOf_two_smul];
      rfl

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