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Theorem cnlnadjlem1 29502
Description: Lemma for cnlnadji 29511 (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 6448 . . 3 (𝑔 = 𝐴 → (𝑇𝑔) = (𝑇𝐴))
21oveq1d 6939 . 2 (𝑔 = 𝐴 → ((𝑇𝑔) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
3 cnlnadjlem.3 . 2 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))
4 ovex 6956 . 2 ((𝑇𝐴) ·ih 𝑦) ∈ V
52, 3, 4fvmpt 6544 1 (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cmpt 4967  cfv 6137  (class class class)co 6924  chba 28352   ·ih csp 28355  ContOpccop 28379  LinOpclo 28380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927
This theorem is referenced by:  cnlnadjlem2  29503  cnlnadjlem3  29504  cnlnadjlem5  29506
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