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Theorem nmorepnf 30747

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 2729	. . . . 5 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
3	1, 2	nmosetre 30743	. . . 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 2729	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
6		eqid 2729	. . . . . 6 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
7	4, 5, 6	nmosetn0 30744	. . . . 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 4301	. . . 4 ⊢ (𝑈 ∈ NrmCVec → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((norm_CV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑊)‘(𝑇‘𝑧)))} ≠ ∅)
9		supxrre2 13267	. . . 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 597	. . 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 1114	. 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 30742	. . 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 2813	. 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 2984	. 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 1086 = wceq 1540 ∈ wcel 2109 {cab 2707 ≠ wne 2925 ∃wrex 3053 ⊆ wss 3911 ∅c0 4292 class class class wbr 5102 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 supcsup 9367 ℝcr 11043 1c1 11045 +∞cpnf 11181 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 NrmCVeccnv 30563 BaseSetcba 30565 0_veccn0v 30567 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-pss 3931 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-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 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-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 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-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-grpo 30472 df-gid 30473 df-ginv 30474 df-ablo 30524 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: nmoreltpnf 30748 nmogtmnf 30749 nmounbi 30755

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