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Theorem nmorepnf 30856

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 2737	. . . . 5 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
3	1, 2	nmosetre 30852	. . . 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 2737	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
6		eqid 2737	. . . . . 6 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
7	4, 5, 6	nmosetn0 30853	. . . . 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 4296	. . . 4 ⊢ (𝑈 ∈ NrmCVec → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ≠ ∅)
9		supxrre2 13258	. . . 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 30851	. . 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 2822	. 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 2992	. 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 2715 ≠ wne 2933 ∃wrex 3062 ⊆ wss 3903 ∅c0 4287 class class class wbr 5100 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 supcsup 9355 ℝcr 11037 1c1 11039 +∞cpnf 11175 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 NrmCVeccnv 30672 BaseSetcba 30674 0_veccn0v 30676 norm_CVcnmcv 30678 normOp_OLD cnmoo 30829

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-grpo 30581 df-gid 30582 df-ginv 30583 df-ablo 30633 df-vc 30647 df-nv 30680 df-va 30683 df-ba 30684 df-sm 30685 df-0v 30686 df-nmcv 30688 df-nmoo 30833

This theorem is referenced by: nmoreltpnf 30857 nmogtmnf 30858 nmounbi 30864

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