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Theorem nmoxr 31118

Description: The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
nmoxr	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)

Proof of Theorem nmoxr Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoxr.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
2		nmoxr.2	. . 3 ⊢ 𝑌 = (BaseSet‘𝑊)
3		eqid 2763	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
4		eqid 2763	. . 3 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
5		nmoxr.3	. . 3 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
6	1, 2, 3, 4, 5	nmooval 31115	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^*, < ))
7	2, 4	nmosetre 31116	. . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ)
8		ressxr 11248	. . . . 5 ⊢ ℝ ⊆ ℝ^*
9	7, 8	sstrdi 3949	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ^*)
10		supxrcl 13336	. . . 4 ⊢ ({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ^* → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ∈ ℝ^)
11	9, 10	syl 18	. . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ∈ ℝ^)
12	11	3adant1 1148	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ∈ ℝ^)
13	6, 12	eqeltrd 2863	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 {cab 2741 ∃wrex 3089 ⊆ wss 3905 class class class wbr 5109 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 supcsup 9396 ℝcr 11094 1c1 11096 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 NrmCVeccnv 30936 BaseSetcba 30938 norm_CVcnmcv 30942 normOp_OLD cnmoo 31093

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-po 5569 df-so 5570 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 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 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-vc 30911 df-nv 30944 df-va 30947 df-ba 30948 df-sm 30949 df-0v 30950 df-nmcv 30952 df-nmoo 31097

This theorem is referenced by: nmooge0 31119 nmoreltpnf 31121 nmobndi 31127 nmblore 31138 nmlnogt0 31149 isblo3i 31153 ubthlem3 31224

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