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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 29911 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) |
5 | 4 | necon3bid 2984 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 𝑇 ≠ 𝑍)) |
6 | eqid 2731 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
7 | eqid 2731 | . . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
8 | 6, 7, 3 | lnof 29871 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) |
9 | 6, 7, 1 | nmoxr 29882 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ*) |
10 | 6, 7, 1 | nmooge0 29883 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇)) |
11 | 0xr 11243 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
12 | xrlttri2 13103 | . . . . . . 7 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) | |
13 | 11, 12 | mpan2 689 | . . . . . 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 11261 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ (𝑁‘𝑇) ∈ ℝ*) → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0)) | |
16 | 11, 15 | mpan 688 | . . . . . . 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 935 | . . . . . 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 281 | . . . 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 1409 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇))) |
23 | 5, 22 | bitr3d 280 | 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 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 class class class wbr 5141 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 0cc0 11092 ℝ*cxr 11229 < clt 11230 ≤ cle 11231 NrmCVeccnv 29700 BaseSetcba 29702 LnOp clno 29856 normOpOLD cnmoo 29857 0op c0o 29859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-grpo 29609 df-gid 29610 df-ginv 29611 df-ablo 29661 df-vc 29675 df-nv 29708 df-va 29711 df-ba 29712 df-sm 29713 df-0v 29714 df-nmcv 29716 df-lno 29860 df-nmoo 29861 df-0o 29863 |
This theorem is referenced by: blocni 29921 |
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