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Mirrors > Home > MPE Home > Th. List > nmlnogt0 | Structured version Visualization version GIF version |
Description: The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmlnogt0.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
nmlnogt0.0 | ⊢ 𝑍 = (𝑈 0op 𝑊) |
nmlnogt0.7 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
Ref | Expression |
---|---|
nmlnogt0 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmlnogt0.3 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
2 | nmlnogt0.0 | . . . 4 ⊢ 𝑍 = (𝑈 0op 𝑊) | |
3 | nmlnogt0.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
4 | 1, 2, 3 | nmlno0 28499 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) |
5 | 4 | necon3bid 3057 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 𝑇 ≠ 𝑍)) |
6 | eqid 2818 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
7 | eqid 2818 | . . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
8 | 6, 7, 3 | lnof 28459 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) |
9 | 6, 7, 1 | nmoxr 28470 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ*) |
10 | 6, 7, 1 | nmooge0 28471 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇)) |
11 | 0xr 10676 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
12 | xrlttri2 12523 | . . . . . . 7 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) | |
13 | 11, 12 | mpan2 687 | . . . . . 6 ⊢ ((𝑁‘𝑇) ∈ ℝ* → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) |
14 | 13 | adantr 481 | . . . . 5 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) |
15 | xrlenlt 10694 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ (𝑁‘𝑇) ∈ ℝ*) → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0)) | |
16 | 11, 15 | mpan 686 | . . . . . . 7 ⊢ ((𝑁‘𝑇) ∈ ℝ* → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0)) |
17 | 16 | biimpa 477 | . . . . . 6 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝑇)) → ¬ (𝑁‘𝑇) < 0) |
18 | biorf 930 | . . . . . 6 ⊢ (¬ (𝑁‘𝑇) < 0 → (0 < (𝑁‘𝑇) ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝑇)) → (0 < (𝑁‘𝑇) ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) |
20 | 14, 19 | bitr4d 283 | . . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇))) |
21 | 9, 10, 20 | syl2anc 584 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇))) |
22 | 8, 21 | syld3an3 1401 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇))) |
23 | 5, 22 | bitr3d 282 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 NrmCVeccnv 28288 BaseSetcba 28290 LnOp clno 28444 normOpOLD cnmoo 28445 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-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-0o 28451 |
This theorem is referenced by: blocni 28509 |
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