Theorem cnlnadjlem1 32389

Description: Lemma for cnlnadji 32398 (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 ↦ ((𝑇‘𝑔) ·ih 𝑦))

Assertion

Ref	Expression
cnlnadjlem1	⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦))

Distinct variable groups:   𝑦,𝑔,𝐴   𝑇,𝑔,𝑦

Allowed substitution hints:   𝐺(𝑦,𝑔)

Proof of Theorem cnlnadjlem1

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴))
2	1	oveq1d 7429	. 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦))
3		cnlnadjlem.3	. 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))
4		ovex 7447	. 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V
5	2, 3, 4	fvmpt 6993	1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2150   ↦ cmpt 5197  ‘cfv 6540  (class class class)co 7414   ℋchba 31241   ·_ih csp 31244  ContOpccop 31268  LinOpclo 31269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7417

This theorem is referenced by:  cnlnadjlem2  32390  cnlnadjlem3  32391  cnlnadjlem5  32393