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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 30824 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) |
5 | 4 | necon3bid 2983 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 𝑇 ≠ 𝑍)) |
6 | eqid 2735 | . . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
7 | eqid 2735 | . . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
8 | 6, 7, 3 | lnof 30784 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) |
9 | 6, 7, 1 | nmoxr 30795 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ*) |
10 | 6, 7, 1 | nmooge0 30796 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇)) |
11 | 0xr 11306 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
12 | xrlttri2 13181 | . . . . . . 7 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) | |
13 | 11, 12 | mpan2 691 | . . . . . 6 ⊢ ((𝑁‘𝑇) ∈ ℝ* → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) |
14 | 13 | adantr 480 | . . . . 5 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇)))) |
15 | xrlenlt 11324 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ (𝑁‘𝑇) ∈ ℝ*) → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0)) | |
16 | 11, 15 | mpan 690 | . . . . . . 7 ⊢ ((𝑁‘𝑇) ∈ ℝ* → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0)) |
17 | 16 | biimpa 476 | . . . . . 6 ⊢ (((𝑁‘𝑇) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝑇)) → ¬ (𝑁‘𝑇) < 0) |
18 | biorf 936 | . . . . . 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 282 | . . . 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 1408 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇))) |
23 | 5, 22 | bitr3d 281 | 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 NrmCVeccnv 30613 BaseSetcba 30615 LnOp clno 30769 normOpOLD cnmoo 30770 0op c0o 30772 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-grpo 30522 df-gid 30523 df-ginv 30524 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-nmcv 30629 df-lno 30773 df-nmoo 30774 df-0o 30776 |
This theorem is referenced by: blocni 30834 |
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