Theorem cnlnadjlem1 32696

Description: Lemma for cnlnadji 32705 (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 6905	. . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴))
2	1	oveq1d 7455	. 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦))
3		cnlnadjlem.3	. 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))
4		ovex 7473	. 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V
5	2, 3, 4	fvmpt 7013	1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ↦ cmpt 5189  ‘cfv 6555  (class class class)co 7440   ℋchba 31548   ·_ih csp 31551  ContOpccop 31575  LinOpclo 31576

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-ov 7443

This theorem is used by:  cnlnadjlem2  32697  cnlnadjlem3  32698  cnlnadjlem5  32700