Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > cnlnadjlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 29847 (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 6664 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴)) | |
2 | 1 | oveq1d 7165 | . 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦)) |
3 | cnlnadjlem.3 | . 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) | |
4 | ovex 7183 | . 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V | |
5 | 2, 3, 4 | fvmpt 6762 | 1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ℋchba 28690 ·ih csp 28693 ContOpccop 28717 LinOpclo 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 |
This theorem is referenced by: cnlnadjlem2 29839 cnlnadjlem3 29840 cnlnadjlem5 29842 |
Copyright terms: Public domain | W3C validator |