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Theorem nmorepnf 30839

Description: The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-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
nmorepnf	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))

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

Step	Hyp	Ref	Expression
1		nmoxr.2	. . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊)
2		eqid 2736	. . . . 5 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
3	1, 2	nmosetre 30835	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ)
4		nmoxr.1	. . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈)
5		eqid 2736	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
6		eqid 2736	. . . . . 6 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
7	4, 5, 6	nmosetn0 30836	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ((norm_CV‘𝑊)‘(𝑇‘(0_vec‘𝑈))) ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))})
8	7	ne0d 4282	. . . 4 ⊢ (𝑈 ∈ NrmCVec → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ≠ ∅)
9		supxrre2 13283	. . . 4 ⊢ (({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ ∧ {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ≠ ∅) → (sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ≠ +∞))
10	3, 8, 9	syl2anr 598	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌)) → (sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ≠ +∞))
11	10	3impb 1115	. 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 ↔ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^, < ) ≠ +∞))
12		nmoxr.3	. . . 4 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
13	4, 1, 6, 2, 12	nmooval 30834	. . 3 ⊢ ((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^*, < ))
14	13	eleq1d 2821	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))}, ℝ^*, < ) ∈ ℝ))
15	13	neeq1d 2991	. 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^*, < ) ≠ +∞))
16	11, 14, 15	3bitr4d 311	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cab 2714 ≠ wne 2932 ∃wrex 3061 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 supcsup 9353 ℝcr 11037 1c1 11039 +∞cpnf 11176 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 NrmCVeccnv 30655 BaseSetcba 30657 0_veccn0v 30659 norm_CVcnmcv 30661 normOp_OLD cnmoo 30812

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 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 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-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-grpo 30564 df-gid 30565 df-ginv 30566 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-nmcv 30671 df-nmoo 30816

This theorem is referenced by: nmoreltpnf 30840 nmogtmnf 30841 nmounbi 30847

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