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Theorem nmlnogt0 28224

Description: The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmlnogt0.3	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmlnogt0.0	⊢ 𝑍 = (𝑈 0_op 𝑊)
nmlnogt0.7	⊢ 𝐿 = (𝑈 LnOp 𝑊)

**Assertion**
Ref	Expression
nmlnogt0	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇)))

**Proof of Theorem nmlnogt0**
Step	Hyp	Ref	Expression
1		nmlnogt0.3	. . . 4 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
2		nmlnogt0.0	. . . 4 ⊢ 𝑍 = (𝑈 0_op 𝑊)
3		nmlnogt0.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
4	1, 2, 3	nmlno0 28222	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))
5	4	necon3bid 3012	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 𝑇 ≠ 𝑍))
6		eqid 2777	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
7		eqid 2777	. . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
8	6, 7, 3	lnof 28182	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
9	6, 7, 1	nmoxr 28193	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ^*)
10	6, 7, 1	nmooge0 28194	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇))
11		0xr 10423	. . . . . . 7 ⊢ 0 ∈ ℝ^*
12		xrlttri2 12285	. . . . . . 7 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
13	11, 12	mpan2 681	. . . . . 6 ⊢ ((𝑁‘𝑇) ∈ ℝ^* → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
14	13	adantr 474	. . . . 5 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
15		xrlenlt 10442	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (𝑁‘𝑇) ∈ ℝ^*) → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0))
16	11, 15	mpan 680	. . . . . . 7 ⊢ ((𝑁‘𝑇) ∈ ℝ^* → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0))
17	16	biimpa 470	. . . . . 6 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → ¬ (𝑁‘𝑇) < 0)
18		biorf 923	. . . . . 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 274	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
21	9, 10, 20	syl2anc 579	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
22	8, 21	syld3an3 1477	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
23	5, 22	bitr3d 273	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 198 ∧ wa 386 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 0cc0 10272 ℝ^*cxr 10410 < clt 10411 ≤ cle 10412 NrmCVeccnv 28011 BaseSetcba 28013 LnOp clno 28167 normOp_OLD cnmoo 28168 0_op c0o 28170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-grpo 27920 df-gid 27921 df-ginv 27922 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-nmcv 28027 df-lno 28171 df-nmoo 28172 df-0o 28174

This theorem is referenced by: blocni 28232

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