Theorem cnlnadjlem1 32430

Description: Lemma for cnlnadji 32439 (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 6881	. . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴))
2	1	oveq1d 7427	. 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦))
3		cnlnadjlem.3	. 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))
4		ovex 7445	. 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V
5	2, 3, 4	fvmpt 6989	1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7412   ℋchba 31282   ·_ih csp 31285  ContOpccop 31309  LinOpclo 31310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415

This theorem is used by:  cnlnadjlem2  32431  cnlnadjlem3  32432  cnlnadjlem5  32434