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Theorem cnlnadjlem1 28109
Description: Lemma for cnlnadji 28118 (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 ∈ ConOp
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 5986 . . 3 (𝑔 = 𝐴 → (𝑇𝑔) = (𝑇𝐴))
21oveq1d 6440 . 2 (𝑔 = 𝐴 → ((𝑇𝑔) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
3 cnlnadjlem.3 . 2 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))
4 ovex 6453 . 2 ((𝑇𝐴) ·ih 𝑦) ∈ V
52, 3, 4fvmpt 6074 1 (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  cmpt 4541  cfv 5689  (class class class)co 6425  chil 26959   ·ih csp 26962  ConOpccop 26986  LinOpclo 26987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6428
This theorem is referenced by:  cnlnadjlem2  28110  cnlnadjlem3  28111  cnlnadjlem5  28113
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