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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 | β’ π = (BaseSetβπ) |
nmoxr.2 | β’ π = (BaseSetβπ) |
nmoxr.3 | β’ π = (π normOpOLD π) |
Ref | Expression |
---|---|
nmoxr | β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β (πβπ) β β*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoxr.1 | . . 3 β’ π = (BaseSetβπ) | |
2 | nmoxr.2 | . . 3 β’ π = (BaseSetβπ) | |
3 | eqid 2728 | . . 3 β’ (normCVβπ) = (normCVβπ) | |
4 | eqid 2728 | . . 3 β’ (normCVβπ) = (normCVβπ) | |
5 | nmoxr.3 | . . 3 β’ π = (π normOpOLD π) | |
6 | 1, 2, 3, 4, 5 | nmooval 30572 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β (πβπ) = sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < )) |
7 | 2, 4 | nmosetre 30573 | . . . . 5 β’ ((π β NrmCVec β§ π:πβΆπ) β {π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))} β β) |
8 | ressxr 11288 | . . . . 5 β’ β β β* | |
9 | 7, 8 | sstrdi 3992 | . . . 4 β’ ((π β NrmCVec β§ π:πβΆπ) β {π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))} β β*) |
10 | supxrcl 13326 | . . . 4 β’ ({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))} β β* β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β β*) | |
11 | 9, 10 | syl 17 | . . 3 β’ ((π β NrmCVec β§ π:πβΆπ) β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β β*) |
12 | 11 | 3adant1 1128 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β β*) |
13 | 6, 12 | eqeltrd 2829 | 1 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β (πβπ) β β*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 {cab 2705 βwrex 3067 β wss 3947 class class class wbr 5148 βΆwf 6544 βcfv 6548 (class class class)co 7420 supcsup 9463 βcr 11137 1c1 11139 β*cxr 11277 < clt 11278 β€ cle 11279 NrmCVeccnv 30393 BaseSetcba 30395 normCVcnmcv 30399 normOpOLD cnmoo 30550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-vc 30368 df-nv 30401 df-va 30404 df-ba 30405 df-sm 30406 df-0v 30407 df-nmcv 30409 df-nmoo 30554 |
This theorem is referenced by: nmooge0 30576 nmoreltpnf 30578 nmobndi 30584 nmblore 30595 nmlnogt0 30606 isblo3i 30610 ubthlem3 30681 |
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