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Theorem cnlnadjlem1 31870
Description: Lemma for cnlnadji 31879 (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.)
Hypotheses
Ref Expression
cnlnadjlem.1 𝑇 ∈ LinOp
cnlnadjlem.2 𝑇 ∈ ContOp
cnlnadjlem.3 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))
Assertion
Ref Expression
cnlnadjlem1 (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))
Distinct variable groups:   𝑦,𝑔,𝐴   𝑇,𝑔,𝑦
Allowed substitution hints:   𝐺(𝑦,𝑔)

Proof of Theorem cnlnadjlem1
StepHypRef Expression
1 fveq2 6891 . . 3 (𝑔 = 𝐴 → (𝑇𝑔) = (𝑇𝐴))
21oveq1d 7429 . 2 (𝑔 = 𝐴 → ((𝑇𝑔) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
3 cnlnadjlem.3 . 2 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))
4 ovex 7447 . 2 ((𝑇𝐴) ·ih 𝑦) ∈ V
52, 3, 4fvmpt 6999 1 (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cmpt 5225  cfv 6542  (class class class)co 7414  chba 30722   ·ih csp 30725  ContOpccop 30749  LinOpclo 30750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417
This theorem is referenced by:  cnlnadjlem2  31871  cnlnadjlem3  31872  cnlnadjlem5  31874
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