![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdvbaselfl | Structured version Visualization version GIF version |
Description: A vector in the base set of the closed kernel dual space is a functional. (Contributed by NM, 28-Mar-2015.) |
Ref | Expression |
---|---|
lcdvbasess.h | β’ π» = (LHypβπΎ) |
lcdvbasess.c | β’ πΆ = ((LCDualβπΎ)βπ) |
lcdvbasess.v | β’ π = (BaseβπΆ) |
lcdvbasess.u | β’ π = ((DVecHβπΎ)βπ) |
lcdvbasess.f | β’ πΉ = (LFnlβπ) |
lcdvbasess.k | β’ (π β (πΎ β HL β§ π β π»)) |
lcdvbaselfl.x | β’ (π β π β π) |
Ref | Expression |
---|---|
lcdvbaselfl | β’ (π β π β πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdvbasess.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | lcdvbasess.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
3 | lcdvbasess.v | . . 3 β’ π = (BaseβπΆ) | |
4 | lcdvbasess.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
5 | lcdvbasess.f | . . 3 β’ πΉ = (LFnlβπ) | |
6 | lcdvbasess.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
7 | 1, 2, 3, 4, 5, 6 | lcdvbasess 41122 | . 2 β’ (π β π β πΉ) |
8 | lcdvbaselfl.x | . 2 β’ (π β π β π) | |
9 | 7, 8 | sseldd 3973 | 1 β’ (π β π β πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6542 Basecbs 17177 LFnlclfn 38584 HLchlt 38877 LHypclh 39512 DVecHcdvh 40606 LCDualclcd 41114 |
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 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38480 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lvec 20990 df-ldual 38651 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 df-tendo 40283 df-edring 40285 df-dvech 40607 df-lcdual 41115 |
This theorem is referenced by: lcdvbasecl 41124 lcdvaddval 41126 lcdvsval 41132 lcdlkreqN 41150 lcdlkreq2N 41151 mapd0 41193 hvmaplfl 41295 hdmapln1 41434 hdmaplna1 41435 hdmaplns1 41436 hdmaplnm1 41437 hdmaplkr 41441 hdmapellkr 41442 hdmapip0 41443 hdmapinvlem1 41446 |
Copyright terms: Public domain | W3C validator |