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Mirrors > Home > MPE Home > Th. List > nmblolbi | Structured version Visualization version GIF version |
Description: A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmblolbi.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmblolbi.4 | ⊢ 𝐿 = (normCV‘𝑈) |
nmblolbi.5 | ⊢ 𝑀 = (normCV‘𝑊) |
nmblolbi.6 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
nmblolbi.7 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
nmblolbi.u | ⊢ 𝑈 ∈ NrmCVec |
nmblolbi.w | ⊢ 𝑊 ∈ NrmCVec |
Ref | Expression |
---|---|
nmblolbi | ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6896 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) → (𝑇‘𝐴) = (if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))‘𝐴)) | |
2 | 1 | fveq2d 6901 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) → (𝑀‘(𝑇‘𝐴)) = (𝑀‘(if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))‘𝐴))) |
3 | fveq2 6897 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) → (𝑁‘𝑇) = (𝑁‘if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)))) | |
4 | 3 | oveq1d 7435 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) → ((𝑁‘𝑇) · (𝐿‘𝐴)) = ((𝑁‘if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))) · (𝐿‘𝐴))) |
5 | 2, 4 | breq12d 5161 | . . . 4 ⊢ (𝑇 = if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) → ((𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)) ↔ (𝑀‘(if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))‘𝐴)) ≤ ((𝑁‘if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))) · (𝐿‘𝐴)))) |
6 | 5 | imbi2d 340 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) → ((𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) ↔ (𝐴 ∈ 𝑋 → (𝑀‘(if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))‘𝐴)) ≤ ((𝑁‘if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))) · (𝐿‘𝐴))))) |
7 | nmblolbi.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | nmblolbi.4 | . . . 4 ⊢ 𝐿 = (normCV‘𝑈) | |
9 | nmblolbi.5 | . . . 4 ⊢ 𝑀 = (normCV‘𝑊) | |
10 | nmblolbi.6 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
11 | nmblolbi.7 | . . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
12 | nmblolbi.u | . . . 4 ⊢ 𝑈 ∈ NrmCVec | |
13 | nmblolbi.w | . . . 4 ⊢ 𝑊 ∈ NrmCVec | |
14 | eqid 2728 | . . . . . . 7 ⊢ (𝑈 0op 𝑊) = (𝑈 0op 𝑊) | |
15 | 14, 11 | 0blo 30615 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) ∈ 𝐵) |
16 | 12, 13, 15 | mp2an 691 | . . . . 5 ⊢ (𝑈 0op 𝑊) ∈ 𝐵 |
17 | 16 | elimel 4598 | . . . 4 ⊢ if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊)) ∈ 𝐵 |
18 | 7, 8, 9, 10, 11, 12, 13, 17 | nmblolbii 30622 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))‘𝐴)) ≤ ((𝑁‘if(𝑇 ∈ 𝐵, 𝑇, (𝑈 0op 𝑊))) · (𝐿‘𝐴))) |
19 | 6, 18 | dedth 4587 | . 2 ⊢ (𝑇 ∈ 𝐵 → (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))) |
20 | 19 | imp 406 | 1 ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ifcif 4529 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 · cmul 11144 ≤ cle 11280 NrmCVeccnv 30407 BaseSetcba 30409 normCVcnmcv 30413 normOpOLD cnmoo 30564 BLnOp cblo 30565 0op c0o 30566 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-grpo 30316 df-gid 30317 df-ginv 30318 df-ablo 30368 df-vc 30382 df-nv 30415 df-va 30418 df-ba 30419 df-sm 30420 df-0v 30421 df-nmcv 30423 df-lno 30567 df-nmoo 30568 df-blo 30569 df-0o 30570 |
This theorem is referenced by: isblo3i 30624 blometi 30626 ubthlem3 30695 htthlem 30740 |
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