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Mirrors > Home > HSE Home > Th. List > cnlnadjlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 29511 (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.) |
Ref | Expression |
---|---|
cnlnadjlem.1 | ⊢ 𝑇 ∈ LinOp |
cnlnadjlem.2 | ⊢ 𝑇 ∈ ContOp |
cnlnadjlem.3 | ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) |
Ref | Expression |
---|---|
cnlnadjlem1 | ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6448 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴)) | |
2 | 1 | oveq1d 6939 | . 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦)) |
3 | cnlnadjlem.3 | . 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) | |
4 | ovex 6956 | . 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V | |
5 | 2, 3, 4 | fvmpt 6544 | 1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ℋchba 28352 ·ih csp 28355 ContOpccop 28379 LinOpclo 28380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-ov 6927 |
This theorem is referenced by: cnlnadjlem2 29503 cnlnadjlem3 29504 cnlnadjlem5 29506 |
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