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Theorem cnlnadjlem1 30425
Description: Lemma for cnlnadji 30434 (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 6771 . . 3 (𝑔 = 𝐴 → (𝑇𝑔) = (𝑇𝐴))
21oveq1d 7286 . 2 (𝑔 = 𝐴 → ((𝑇𝑔) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
3 cnlnadjlem.3 . 2 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))
4 ovex 7304 . 2 ((𝑇𝐴) ·ih 𝑦) ∈ V
52, 3, 4fvmpt 6872 1 (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  cmpt 5162  cfv 6432  (class class class)co 7271  chba 29277   ·ih csp 29280  ContOpccop 29304  LinOpclo 29305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7274
This theorem is referenced by:  cnlnadjlem2  30426  cnlnadjlem3  30427  cnlnadjlem5  30429
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