Theorem cnlnadjlem1 32272

Description: Lemma for cnlnadji 32281 (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 6869	. . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴))
2	1	oveq1d 7413	. 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦))
3		cnlnadjlem.3	. 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))
4		ovex 7431	. 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V
5	2, 3, 4	fvmpt 6977	1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦))

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   ↦ cmpt 5183  ‘cfv 6523  (class class class)co 7398   ℋchba 31124   ·_ih csp 31127  ContOpccop 31151  LinOpclo 31152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401

This theorem is referenced by:  cnlnadjlem2  32273  cnlnadjlem3  32274  cnlnadjlem5  32276