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Mirrors > Home > MPE Home > Th. List > nmorepnf | Structured version Visualization version GIF version |
Description: The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-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 |
---|---|
nmorepnf | β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β β β (πβπ) β +β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoxr.2 | . . . . 5 β’ π = (BaseSetβπ) | |
2 | eqid 2728 | . . . . 5 β’ (normCVβπ) = (normCVβπ) | |
3 | 1, 2 | nmosetre 30594 | . . . 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) |
4 | nmoxr.1 | . . . . . 6 β’ π = (BaseSetβπ) | |
5 | eqid 2728 | . . . . . 6 β’ (0vecβπ) = (0vecβπ) | |
6 | eqid 2728 | . . . . . 6 β’ (normCVβπ) = (normCVβπ) | |
7 | 4, 5, 6 | nmosetn0 30595 | . . . . 5 β’ (π β NrmCVec β ((normCVβπ)β(πβ(0vecβπ))) β {π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}) |
8 | 7 | ne0d 4339 | . . . 4 β’ (π β NrmCVec β {π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))} β β ) |
9 | supxrre2 13350 | . . . 4 β’ (({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))} β β 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 β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β +β)) | |
10 | 3, 8, 9 | syl2anr 595 | . . 3 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 β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β +β)) |
11 | 10 | 3impb 1112 | . 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 β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β +β)) |
12 | nmoxr.3 | . . . 4 β’ π = (π normOpOLD π) | |
13 | 4, 1, 6, 2, 12 | nmooval 30593 | . . 3 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*, < )) |
14 | 13 | eleq1d 2814 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β β β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β β)) |
15 | 13 | neeq1d 2997 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β +β β sup({π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))}, β*, < ) β +β)) |
16 | 11, 14, 15 | 3bitr4d 310 | 1 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β β β (πβπ) β +β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {cab 2705 β wne 2937 βwrex 3067 β wss 3949 β c0 4326 class class class wbr 5152 βΆwf 6549 βcfv 6553 (class class class)co 7426 supcsup 9471 βcr 11145 1c1 11147 +βcpnf 11283 β*cxr 11285 < clt 11286 β€ cle 11287 NrmCVeccnv 30414 BaseSetcba 30416 0veccn0v 30418 normCVcnmcv 30420 normOpOLD cnmoo 30571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-grpo 30323 df-gid 30324 df-ginv 30325 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-nmcv 30430 df-nmoo 30575 |
This theorem is referenced by: nmoreltpnf 30599 nmogtmnf 30600 nmounbi 30606 |
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