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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 2726 | . . . . 5 β’ (normCVβπ) = (normCVβπ) | |
3 | 1, 2 | nmosetre 30521 | . . . 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 2726 | . . . . . 6 β’ (0vecβπ) = (0vecβπ) | |
6 | eqid 2726 | . . . . . 6 β’ (normCVβπ) = (normCVβπ) | |
7 | 4, 5, 6 | nmosetn0 30522 | . . . . 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 4330 | . . . 4 β’ (π β NrmCVec β {π₯ β£ βπ§ β π (((normCVβπ)βπ§) β€ 1 β§ π₯ = ((normCVβπ)β(πβπ§)))} β β ) |
9 | supxrre2 13313 | . . . 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 596 | . . 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 30520 | . . 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 2812 | . 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 2994 | . 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 311 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {cab 2703 β wne 2934 βwrex 3064 β wss 3943 β c0 4317 class class class wbr 5141 βΆwf 6532 βcfv 6536 (class class class)co 7404 supcsup 9434 βcr 11108 1c1 11110 +βcpnf 11246 β*cxr 11248 < clt 11249 β€ cle 11250 NrmCVeccnv 30341 BaseSetcba 30343 0veccn0v 30345 normCVcnmcv 30347 normOpOLD cnmoo 30498 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-grpo 30250 df-gid 30251 df-ginv 30252 df-ablo 30302 df-vc 30316 df-nv 30349 df-va 30352 df-ba 30353 df-sm 30354 df-0v 30355 df-nmcv 30357 df-nmoo 30502 |
This theorem is referenced by: nmoreltpnf 30526 nmogtmnf 30527 nmounbi 30533 |
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