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Mirrors > Home > MPE Home > Th. List > blo3i | Structured version Visualization version GIF version |
Description: Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isblo3i.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
isblo3i.m | ⊢ 𝑀 = (normCV‘𝑈) |
isblo3i.n | ⊢ 𝑁 = (normCV‘𝑊) |
isblo3i.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
isblo3i.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
isblo3i.u | ⊢ 𝑈 ∈ NrmCVec |
isblo3i.w | ⊢ 𝑊 ∈ NrmCVec |
Ref | Expression |
---|---|
blo3i | ⊢ ((𝑇 ∈ 𝐿 ∧ 𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7152 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑀‘𝑦)) = (𝐴 · (𝑀‘𝑦))) | |
2 | 1 | breq2d 5069 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦)) ↔ (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦)))) |
3 | 2 | ralbidv 3194 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦)))) |
4 | 3 | rspcev 3620 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦))) |
5 | isblo3i.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | isblo3i.m | . . . . 5 ⊢ 𝑀 = (normCV‘𝑈) | |
7 | isblo3i.n | . . . . 5 ⊢ 𝑁 = (normCV‘𝑊) | |
8 | isblo3i.4 | . . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
9 | isblo3i.5 | . . . . 5 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
10 | isblo3i.u | . . . . 5 ⊢ 𝑈 ∈ NrmCVec | |
11 | isblo3i.w | . . . . 5 ⊢ 𝑊 ∈ NrmCVec | |
12 | 5, 6, 7, 8, 9, 10, 11 | isblo3i 28505 | . . . 4 ⊢ (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦)))) |
13 | 12 | biimpri 229 | . . 3 ⊢ ((𝑇 ∈ 𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵) |
14 | 4, 13 | sylan2 592 | . 2 ⊢ ((𝑇 ∈ 𝐿 ∧ (𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦)))) → 𝑇 ∈ 𝐵) |
15 | 14 | 3impb 1107 | 1 ⊢ ((𝑇 ∈ 𝐿 ∧ 𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 · cmul 10530 ≤ cle 10664 NrmCVeccnv 28288 BaseSetcba 28290 normCVcnmcv 28294 LnOp clno 28444 BLnOp cblo 28446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-grpo 28197 df-gid 28198 df-ginv 28199 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 df-lno 28448 df-nmoo 28449 df-blo 28450 df-0o 28451 |
This theorem is referenced by: ipblnfi 28559 |
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