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Theorem nmlnogt0 30777

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 30775	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))
5	4	necon3bid 2969	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 𝑇 ≠ 𝑍))
6		eqid 2729	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
7		eqid 2729	. . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
8	6, 7, 3	lnof 30735	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
9	6, 7, 1	nmoxr 30746	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ^*)
10	6, 7, 1	nmooge0 30747	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇))
11		0xr 11199	. . . . . . 7 ⊢ 0 ∈ ℝ^*
12		xrlttri2 13080	. . . . . . 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 11217	. . . . . . . 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 1411	. 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 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5102 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 0cc0 11046 ℝ^*cxr 11185 < clt 11186 ≤ cle 11187 NrmCVeccnv 30564 BaseSetcba 30566 LnOp clno 30720 normOp_OLD cnmoo 30721 0_op c0o 30723

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 ax-pre-sup 11124 ax-addf 11125 ax-mulf 11126

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-div 11814 df-nn 12165 df-2 12227 df-3 12228 df-n0 12421 df-z 12508 df-uz 12772 df-rp 12930 df-seq 13945 df-exp 14005 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-grpo 30473 df-gid 30474 df-ginv 30475 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-nmcv 30580 df-lno 30724 df-nmoo 30725 df-0o 30727

This theorem is referenced by: blocni 30785

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