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| Mirrors > Home > MPE Home > Th. List > nmoreltpnf | Structured version Visualization version GIF version | ||
| Description: The norm of any operator is real iff it is less than 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 |
|---|---|
| nmoreltpnf | β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β β β (πβπ) < +β)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nmoxr.1 | . . 3 β’ π = (BaseSetβπ) | |
| 2 | nmoxr.2 | . . 3 β’ π = (BaseSetβπ) | |
| 3 | nmoxr.3 | . . 3 β’ π = (π normOpOLD π) | |
| 4 | 1, 2, 3 | nmorepnf 30622 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β β β (πβπ) β +β)) |
| 5 | 1, 2, 3 | nmoxr 30620 | . . . 4 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β (πβπ) β β*) |
| 6 | nltpnft 13175 | . . . 4 β’ ((πβπ) β β* β ((πβπ) = +β β Β¬ (πβπ) < +β)) | |
| 7 | 5, 6 | syl 17 | . . 3 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) = +β β Β¬ (πβπ) < +β)) |
| 8 | 7 | necon2abid 2973 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) < +β β (πβπ) β +β)) |
| 9 | 4, 8 | bitr4d 281 | 1 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) β β β (πβπ) < +β)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5143 βΆwf 6539 βcfv 6543 (class class class)co 7416 βcr 11137 +βcpnf 11275 β*cxr 11277 < clt 11278 NrmCVeccnv 30438 BaseSetcba 30440 normOpOLD cnmoo 30595 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 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 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 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-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-grpo 30347 df-gid 30348 df-ginv 30349 df-ablo 30399 df-vc 30413 df-nv 30446 df-va 30449 df-ba 30450 df-sm 30451 df-0v 30452 df-nmcv 30454 df-nmoo 30599 |
| This theorem is referenced by: isblo2 30637 |
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