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Theorem nmlnogt0 30856

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 30854	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))
5	4	necon3bid 2974	. 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 30814	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
9	6, 7, 1	nmoxr 30825	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ^*)
10	6, 7, 1	nmooge0 30826	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇))
11		0xr 11181	. . . . . . 7 ⊢ 0 ∈ ℝ^*
12		xrlttri2 13082	. . . . . . 7 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
13	11, 12	mpan2 692	. . . . . 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 11199	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (𝑁‘𝑇) ∈ ℝ^*) → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0))
16	11, 15	mpan 691	. . . . . . 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 937	. . . . . 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 585	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
22	8, 21	syld3an3 1412	. 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 class class class wbr 5074 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 0cc0 11027 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 NrmCVeccnv 30643 BaseSetcba 30645 LnOp clno 30799 normOp_OLD cnmoo 30800 0_op c0o 30802

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8632 df-map 8764 df-en 8883 df-dom 8884 df-sdom 8885 df-sup 9344 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-z 12514 df-uz 12778 df-rp 12932 df-seq 13953 df-exp 14013 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-grpo 30552 df-gid 30553 df-ginv 30554 df-ablo 30604 df-vc 30618 df-nv 30651 df-va 30654 df-ba 30655 df-sm 30656 df-0v 30657 df-nmcv 30659 df-lno 30803 df-nmoo 30804 df-0o 30806

This theorem is referenced by: blocni 30864

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