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Mirrors > Home > MPE Home > Th. List > blocn | Structured version Visualization version GIF version |
Description: A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
blocn.8 | ⊢ 𝐶 = (IndMet‘𝑈) |
blocn.d | ⊢ 𝐷 = (IndMet‘𝑊) |
blocn.j | ⊢ 𝐽 = (MetOpen‘𝐶) |
blocn.k | ⊢ 𝐾 = (MetOpen‘𝐷) |
blocn.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
blocn.u | ⊢ 𝑈 ∈ NrmCVec |
blocn.w | ⊢ 𝑊 ∈ NrmCVec |
blocn.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
Ref | Expression |
---|---|
blocn | ⊢ (𝑇 ∈ 𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2897 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ (𝐽 Cn 𝐾))) | |
2 | eleq1 2897 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) → (𝑇 ∈ 𝐵 ↔ if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ 𝐵)) | |
3 | 1, 2 | bibi12d 347 | . 2 ⊢ (𝑇 = if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) → ((𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵) ↔ (if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ (𝐽 Cn 𝐾) ↔ if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ 𝐵))) |
4 | blocn.8 | . . 3 ⊢ 𝐶 = (IndMet‘𝑈) | |
5 | blocn.d | . . 3 ⊢ 𝐷 = (IndMet‘𝑊) | |
6 | blocn.j | . . 3 ⊢ 𝐽 = (MetOpen‘𝐶) | |
7 | blocn.k | . . 3 ⊢ 𝐾 = (MetOpen‘𝐷) | |
8 | blocn.4 | . . 3 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
9 | blocn.5 | . . 3 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
10 | blocn.u | . . 3 ⊢ 𝑈 ∈ NrmCVec | |
11 | blocn.w | . . 3 ⊢ 𝑊 ∈ NrmCVec | |
12 | eqid 2818 | . . . . . 6 ⊢ (𝑈 0op 𝑊) = (𝑈 0op 𝑊) | |
13 | 12, 8 | 0lno 28494 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) ∈ 𝐿) |
14 | 10, 11, 13 | mp2an 688 | . . . 4 ⊢ (𝑈 0op 𝑊) ∈ 𝐿 |
15 | 14 | elimel 4530 | . . 3 ⊢ if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ 𝐿 |
16 | 4, 5, 6, 7, 8, 9, 10, 11, 15 | blocni 28509 | . 2 ⊢ (if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ (𝐽 Cn 𝐾) ↔ if(𝑇 ∈ 𝐿, 𝑇, (𝑈 0op 𝑊)) ∈ 𝐵) |
17 | 3, 16 | dedth 4519 | 1 ⊢ (𝑇 ∈ 𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ifcif 4463 ‘cfv 6348 (class class class)co 7145 MetOpencmopn 20463 Cn ccn 21760 NrmCVeccnv 28288 IndMetcims 28295 LnOp clno 28444 BLnOp cblo 28446 0op c0o 28447 |
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 ax-addf 10604 ax-mulf 10605 |
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-inf 8895 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-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-cn 21763 df-cnp 21764 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-lno 28448 df-nmoo 28449 df-blo 28450 df-0o 28451 |
This theorem is referenced by: blocn2 28512 |
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