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Mirrors > Home > MPE Home > Th. List > isblo | Structured version Visualization version GIF version |
Description: The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bloval.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
bloval.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
bloval.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
Ref | Expression |
---|---|
isblo | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bloval.3 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
2 | bloval.4 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
3 | bloval.5 | . . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
4 | 1, 2, 3 | bloval 30590 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
5 | 4 | eleq2d 2815 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ 𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})) |
6 | fveq2 6897 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑁‘𝑡) = (𝑁‘𝑇)) | |
7 | 6 | breq1d 5158 | . . 3 ⊢ (𝑡 = 𝑇 → ((𝑁‘𝑡) < +∞ ↔ (𝑁‘𝑇) < +∞)) |
8 | 7 | elrab 3682 | . 2 ⊢ (𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)) |
9 | 5, 8 | bitrdi 287 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 +∞cpnf 11275 < clt 11278 NrmCVeccnv 30393 LnOp clno 30549 normOpOLD cnmoo 30550 BLnOp cblo 30551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-blo 30555 |
This theorem is referenced by: isblo2 30592 bloln 30593 nmblore 30595 isblo3i 30610 htthlem 30726 |
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