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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 28893 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
5 | 4 | eleq2d 2825 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ 𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})) |
6 | fveq2 6738 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑁‘𝑡) = (𝑁‘𝑇)) | |
7 | 6 | breq1d 5079 | . . 3 ⊢ (𝑡 = 𝑇 → ((𝑁‘𝑡) < +∞ ↔ (𝑁‘𝑇) < +∞)) |
8 | 7 | elrab 3617 | . 2 ⊢ (𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)) |
9 | 5, 8 | bitrdi 290 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {crab 3068 class class class wbr 5069 ‘cfv 6400 (class class class)co 7234 +∞cpnf 10893 < clt 10896 NrmCVeccnv 28696 LnOp clno 28852 normOpOLD cnmoo 28853 BLnOp cblo 28854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pr 5338 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-iota 6358 df-fun 6402 df-fv 6408 df-ov 7237 df-oprab 7238 df-mpo 7239 df-blo 28858 |
This theorem is referenced by: isblo2 28895 bloln 28896 nmblore 28898 isblo3i 28913 htthlem 29029 |
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