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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 30530 | . 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 30528 | . . . 4 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β (πβπ) β β*) |
| 6 | nltpnft 13149 | . . . 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 2977 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π:πβΆπ) β ((πβπ) < +β β (πβπ) β +β)) |
| 9 | 4, 8 | bitr4d 282 | 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 2934 class class class wbr 5141 βΆwf 6533 βcfv 6537 (class class class)co 7405 βcr 11111 +βcpnf 11249 β*cxr 11251 < clt 11252 NrmCVeccnv 30346 BaseSetcba 30348 normOpOLD cnmoo 30503 |
| 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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-grpo 30255 df-gid 30256 df-ginv 30257 df-ablo 30307 df-vc 30321 df-nv 30354 df-va 30357 df-ba 30358 df-sm 30359 df-0v 30360 df-nmcv 30362 df-nmoo 30507 |
| This theorem is referenced by: isblo2 30545 |
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