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Mirrors > Home > MPE Home > Th. List > nmoolb | Structured version Visualization version GIF version |
Description: A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoolb.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmoolb.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
nmoolb.l | ⊢ 𝐿 = (normCV‘𝑈) |
nmoolb.m | ⊢ 𝑀 = (normCV‘𝑊) |
nmoolb.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
Ref | Expression |
---|---|
nmoolb | ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoolb.2 | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
2 | nmoolb.m | . . . . . 6 ⊢ 𝑀 = (normCV‘𝑊) | |
3 | 1, 2 | nmosetre 28535 | . . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))} ⊆ ℝ) |
4 | ressxr 10679 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
5 | 3, 4 | sstrdi 3978 | . . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))} ⊆ ℝ*) |
6 | 5 | 3adant1 1126 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))} ⊆ ℝ*) |
7 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝐿‘𝑦) = (𝐿‘𝐴)) | |
8 | 7 | breq1d 5068 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝐿‘𝑦) ≤ 1 ↔ (𝐿‘𝐴) ≤ 1)) |
9 | 2fveq3 6669 | . . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑀‘(𝑇‘𝑦)) = (𝑀‘(𝑇‘𝐴))) | |
10 | 9 | eqeq2d 2832 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦)) ↔ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝐴)))) |
11 | 8, 10 | anbi12d 632 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝐿‘𝑦) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦))) ↔ ((𝐿‘𝐴) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝐴))))) |
12 | eqid 2821 | . . . . . . 7 ⊢ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝐴)) | |
13 | 12 | biantru 532 | . . . . . 6 ⊢ ((𝐿‘𝐴) ≤ 1 ↔ ((𝐿‘𝐴) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝐴)))) |
14 | 11, 13 | syl6bbr 291 | . . . . 5 ⊢ (𝑦 = 𝐴 → (((𝐿‘𝑦) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦))) ↔ (𝐿‘𝐴) ≤ 1)) |
15 | 14 | rspcev 3622 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1) → ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦)))) |
16 | fvex 6677 | . . . . 5 ⊢ (𝑀‘(𝑇‘𝐴)) ∈ V | |
17 | eqeq1 2825 | . . . . . . 7 ⊢ (𝑥 = (𝑀‘(𝑇‘𝐴)) → (𝑥 = (𝑀‘(𝑇‘𝑦)) ↔ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦)))) | |
18 | 17 | anbi2d 630 | . . . . . 6 ⊢ (𝑥 = (𝑀‘(𝑇‘𝐴)) → (((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦))) ↔ ((𝐿‘𝑦) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦))))) |
19 | 18 | rexbidv 3297 | . . . . 5 ⊢ (𝑥 = (𝑀‘(𝑇‘𝐴)) → (∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦))))) |
20 | 16, 19 | elab 3666 | . . . 4 ⊢ ((𝑀‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ (𝑀‘(𝑇‘𝐴)) = (𝑀‘(𝑇‘𝑦)))) |
21 | 15, 20 | sylibr 236 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1) → (𝑀‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))}) |
22 | supxrub 12711 | . . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ (𝑀‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))}) → (𝑀‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))}, ℝ*, < )) | |
23 | 6, 21, 22 | syl2an 597 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))}, ℝ*, < )) |
24 | nmoolb.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
25 | nmoolb.l | . . . 4 ⊢ 𝐿 = (normCV‘𝑈) | |
26 | nmoolb.3 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
27 | 24, 1, 25, 2, 26 | nmooval 28534 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))}, ℝ*, < )) |
28 | 27 | adantr 483 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑦)))}, ℝ*, < )) |
29 | 23, 28 | breqtrrd 5086 | 1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 supcsup 8898 ℝcr 10530 1c1 10532 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 NrmCVeccnv 28355 BaseSetcba 28357 normCVcnmcv 28361 normOpOLD cnmoo 28512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-nmoo 28516 |
This theorem is referenced by: nmblolbii 28570 |
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