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Theorem nmoxr 30745

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 2729	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
4		eqid 2729	. . 3 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
5		nmoxr.3	. . 3 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
6	1, 2, 3, 4, 5	nmooval 30742	. 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 30743	. . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ)
8		ressxr 11194	. . . . 5 ⊢ ℝ ⊆ ℝ^*
9	7, 8	sstrdi 3956	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ^*)
10		supxrcl 13251	. . . 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 17	. . 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 1130	. 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 2828	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {cab 2707 ∃wrex 3053 ⊆ wss 3911 class class class wbr 5102 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 supcsup 9367 ℝcr 11043 1c1 11045 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 NrmCVeccnv 30563 BaseSetcba 30565 norm_CVcnmcv 30569 normOp_OLD cnmoo 30720

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 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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-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-id 5526 df-po 5539 df-so 5540 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-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-1st 7947 df-2nd 7948 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-vc 30538 df-nv 30571 df-va 30574 df-ba 30575 df-sm 30576 df-0v 30577 df-nmcv 30579 df-nmoo 30724

This theorem is referenced by: nmooge0 30746 nmoreltpnf 30748 nmobndi 30754 nmblore 30765 nmlnogt0 30776 isblo3i 30780 ubthlem3 30851

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