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Theorem cnlnadjlem1

Description: Lemma for cnlnadji (Theorem 3.10 of Beran p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional G . (Contributed by NM, 16-Feb-2006) (New usage is discouraged.)

Ref Expression
Hypotheses cnlnadjlem.1 ⊢ T ∈ LinOp
cnlnadjlem.2 ⊢ T ∈ ContOp
cnlnadjlem.3 ⊢ G = g ∈ ℋ ⟼ T ⁡ g ⋅ ih y
Assertion cnlnadjlem1 ⊢ A ∈ ℋ → G ⁡ A = T ⁡ A ⋅ ih y

Proof

Step Hyp Ref Expression
1 cnlnadjlem.1 ⊢ T ∈ LinOp
2 cnlnadjlem.2 ⊢ T ∈ ContOp
3 cnlnadjlem.3 ⊢ G = g ∈ ℋ ⟼ T ⁡ g ⋅ ih y
4 fveq2 ⊢ g = A → T ⁡ g = T ⁡ A
5 4 oveq1d ⊢ g = A → T ⁡ g ⋅ ih y = T ⁡ A ⋅ ih y
6 ovex ⊢ T ⁡ A ⋅ ih y ∈ V
7 5 3 6 fvmpt ⊢ A ∈ ℋ → G ⁡ A = T ⁡ A ⋅ ih y