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Theorem nmlnogt0 31160

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 31158	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))
5	4	necon3bid 3001	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 𝑇 ≠ 𝑍))
6		eqid 2762	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
7		eqid 2762	. . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
8	6, 7, 3	lnof 31118	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
9	6, 7, 1	nmoxr 31129	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁‘𝑇) ∈ ℝ^*)
10	6, 7, 1	nmooge0 31130	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → 0 ≤ (𝑁‘𝑇))
11		0xr 11262	. . . . . . 7 ⊢ 0 ∈ ℝ^*
12		xrlttri2 13173	. . . . . . 7 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
13	11, 12	mpan2 703	. . . . . 6 ⊢ ((𝑁‘𝑇) ∈ ℝ^* → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
14	13	adantr 485	. . . . 5 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
15		xrlenlt 11280	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (𝑁‘𝑇) ∈ ℝ^*) → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0))
16	11, 15	mpan 702	. . . . . . 7 ⊢ ((𝑁‘𝑇) ∈ ℝ^* → (0 ≤ (𝑁‘𝑇) ↔ ¬ (𝑁‘𝑇) < 0))
17	16	biimpa 481	. . . . . 6 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → ¬ (𝑁‘𝑇) < 0)
18		biorf 949	. . . . . 6 ⊢ (¬ (𝑁‘𝑇) < 0 → (0 < (𝑁‘𝑇) ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
19	17, 18	syl 18	. . . . 5 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → (0 < (𝑁‘𝑇) ↔ ((𝑁‘𝑇) < 0 ∨ 0 < (𝑁‘𝑇))))
20	14, 19	bitr4d 285	. . . 4 ⊢ (((𝑁‘𝑇) ∈ ℝ^* ∧ 0 ≤ (𝑁‘𝑇)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
21	9, 10, 20	syl2anc 595	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
22	8, 21	syld3an3 1435	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ≠ 0 ↔ 0 < (𝑁‘𝑇)))
23	5, 22	bitr3d 284	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 class class class wbr 5108 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 0cc0 11106 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 NrmCVeccnv 30947 BaseSetcba 30949 LnOp clno 31103 normOp_OLD cnmoo 31104 0_op c0o 31106

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9400 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-z 12598 df-uz 12869 df-rp 13023 df-seq 14045 df-exp 14105 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-grpo 30856 df-gid 30857 df-ginv 30858 df-ablo 30908 df-vc 30922 df-nv 30955 df-va 30958 df-ba 30959 df-sm 30960 df-0v 30961 df-nmcv 30963 df-lno 31107 df-nmoo 31108 df-0o 31110

This theorem is used by: blocni 31168

W3C validator