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Mirrors > Home > MPE Home > Th. List > nmooge0 | Structured version Visualization version GIF version |
Description: The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoxr.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmoxr.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
nmoxr.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
Ref | Expression |
---|---|
nmooge0 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10676 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ∈ ℝ*) |
3 | simp2 1129 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 𝑊 ∈ NrmCVec) | |
4 | nmoxr.1 | . . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈) | |
5 | eqid 2818 | . . . . . . . 8 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
6 | 4, 5 | nvzcl 28338 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → (0vec‘𝑈) ∈ 𝑋) |
7 | ffvelrn 6841 | . . . . . . 7 ⊢ ((𝑇:𝑋⟶𝑌 ∧ (0vec‘𝑈) ∈ 𝑋) → (𝑇‘(0vec‘𝑈)) ∈ 𝑌) | |
8 | 6, 7 | sylan2 592 | . . . . . 6 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑈 ∈ NrmCVec) → (𝑇‘(0vec‘𝑈)) ∈ 𝑌) |
9 | 8 | ancoms 459 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑇‘(0vec‘𝑈)) ∈ 𝑌) |
10 | 9 | 3adant2 1123 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑇‘(0vec‘𝑈)) ∈ 𝑌) |
11 | nmoxr.2 | . . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊) | |
12 | eqid 2818 | . . . . 5 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
13 | 11, 12 | nvcl 28365 | . . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘(0vec‘𝑈)) ∈ 𝑌) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ∈ ℝ) |
14 | 3, 10, 13 | syl2anc 584 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ∈ ℝ) |
15 | 14 | rexrd 10679 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ∈ ℝ*) |
16 | nmoxr.3 | . . 3 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
17 | 4, 11, 16 | nmoxr 28470 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ*) |
18 | 11, 12 | nvge0 28377 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘(0vec‘𝑈)) ∈ 𝑌) → 0 ≤ ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈)))) |
19 | 3, 10, 18 | syl2anc 584 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈)))) |
20 | 11, 12 | nmosetre 28468 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ) |
21 | ressxr 10673 | . . . . . . 7 ⊢ ℝ ⊆ ℝ* | |
22 | 20, 21 | sstrdi 3976 | . . . . . 6 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ*) |
23 | eqid 2818 | . . . . . . 7 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
24 | 4, 5, 23 | nmosetn0 28469 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}) |
25 | supxrub 12705 | . . . . . 6 ⊢ (({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ* ∧ ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ≤ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < )) | |
26 | 22, 24, 25 | syl2an 595 | . . . . 5 ⊢ (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑈 ∈ NrmCVec) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ≤ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < )) |
27 | 26 | 3impa 1102 | . . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌 ∧ 𝑈 ∈ NrmCVec) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ≤ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < )) |
28 | 27 | 3comr 1117 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ≤ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < )) |
29 | 4, 11, 23, 12, 16 | nmooval 28467 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < )) |
30 | 28, 29 | breqtrrd 5085 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((normCV‘𝑊)‘(𝑇‘(0vec‘𝑈))) ≤ (𝑁‘𝑇)) |
31 | 2, 15, 17, 19, 30 | xrletrd 12543 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {cab 2796 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 supcsup 8892 ℝcr 10524 0cc0 10525 1c1 10526 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 NrmCVeccnv 28288 BaseSetcba 28290 0veccn0v 28292 normCVcnmcv 28294 normOpOLD cnmoo 28445 |
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-nmoo 28449 |
This theorem is referenced by: nmlnogt0 28501 htthlem 28621 |
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