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Mirrors > Home > MPE Home > Th. List > nmoxr | Structured version Visualization version GIF version |
Description: The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoxr.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmoxr.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
nmoxr.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
Ref | Expression |
---|---|
nmoxr | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoxr.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nmoxr.2 | . . 3 ⊢ 𝑌 = (BaseSet‘𝑊) | |
3 | eqid 2738 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
4 | eqid 2738 | . . 3 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
5 | nmoxr.3 | . . 3 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
6 | 1, 2, 3, 4, 5 | nmooval 28698 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < )) |
7 | 2, 4 | nmosetre 28699 | . . . . 5 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ) |
8 | ressxr 10763 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
9 | 7, 8 | sstrdi 3889 | . . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ*) |
10 | supxrcl 12791 | . . . 4 ⊢ ({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))} ⊆ ℝ* → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < ) ∈ ℝ*) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < ) ∈ ℝ*) |
12 | 11 | 3adant1 1131 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 (((normCV‘𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑊)‘(𝑇‘𝑧)))}, ℝ*, < ) ∈ ℝ*) |
13 | 6, 12 | eqeltrd 2833 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {cab 2716 ∃wrex 3054 ⊆ wss 3843 class class class wbr 5030 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 supcsup 8977 ℝcr 10614 1c1 10616 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 NrmCVeccnv 28519 BaseSetcba 28521 normCVcnmcv 28525 normOpOLD cnmoo 28676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-sm 28532 df-0v 28533 df-nmcv 28535 df-nmoo 28680 |
This theorem is referenced by: nmooge0 28702 nmoreltpnf 28704 nmobndi 28710 nmblore 28721 nmlnogt0 28732 isblo3i 28736 ubthlem3 28807 |
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