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Theorem cnlnadjlem1 29848
Description: Lemma for cnlnadji 29857 (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 6659 . . 3 (𝑔 = 𝐴 → (𝑇𝑔) = (𝑇𝐴))
21oveq1d 7161 . 2 (𝑔 = 𝐴 → ((𝑇𝑔) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
3 cnlnadjlem.3 . 2 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))
4 ovex 7179 . 2 ((𝑇𝐴) ·ih 𝑦) ∈ V
52, 3, 4fvmpt 6757 1 (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cmpt 5133  cfv 6344  (class class class)co 7146  chba 28700   ·ih csp 28703  ContOpccop 28727  LinOpclo 28728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149
This theorem is referenced by:  cnlnadjlem2  29849  cnlnadjlem3  29850  cnlnadjlem5  29852
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