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Theorem nmblore 27981

Description: The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmblore.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmblore.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmblore.3	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmblore.5	⊢ 𝐵 = (𝑈 BLnOp 𝑊)

**Assertion**
Ref	Expression
nmblore	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)

**Proof of Theorem nmblore**
Step	Hyp	Ref	Expression
1		nmblore.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		nmblore.2	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
3		nmblore.5	. . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
4	1, 2, 3	blof 27980	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌)
5		nmblore.3	. . . 4 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
6	1, 2, 5	nmogtmnf 27965	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
7	4, 6	syld3an3 1515	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → -∞ < (𝑁‘𝑇))
8		eqid 2771	. . . . 5 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
9	5, 8, 3	isblo 27977	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ (𝑈 LnOp 𝑊) ∧ (𝑁‘𝑇) < +∞)))
10	9	simplbda 487	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) < +∞)
11	10	3impa 1100	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) < +∞)
12	1, 2, 5	nmoxr 27961	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)
13	4, 12	syld3an3 1515	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ^*)
14		xrrebnd 12204	. . 3 ⊢ ((𝑁‘𝑇) ∈ ℝ^* → ((𝑁‘𝑇) ∈ ℝ ↔ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) < +∞)))
15	13, 14	syl 17	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → ((𝑁‘𝑇) ∈ ℝ ↔ (-∞ < (𝑁‘𝑇) ∧ (𝑁‘𝑇) < +∞)))
16	7, 11, 15	mpbir2and 692	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ℝcr 10137 +∞cpnf 10273 -∞cmnf 10274 ℝ^*cxr 10275 < clt 10276 NrmCVeccnv 27779 BaseSetcba 27781 LnOp clno 27935 normOp_OLD cnmoo 27936 BLnOp cblo 27937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-grpo 27687 df-gid 27688 df-ginv 27689 df-ablo 27739 df-vc 27754 df-nv 27787 df-va 27790 df-ba 27791 df-sm 27792 df-0v 27793 df-nmcv 27795 df-lno 27939 df-nmoo 27940 df-blo 27941

This theorem is referenced by: nmblolbii 27994 isblo3i 27996 blocni 28000 htthlem 28114

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