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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhlvec | Structured version Visualization version GIF version |
Description: The full vector space π constructed from a Hilbert lattice πΎ (given a fiducial hyperplane π) is a left module. (Contributed by NM, 23-May-2015.) |
Ref | Expression |
---|---|
dvhlvec.h | β’ π» = (LHypβπΎ) |
dvhlvec.u | β’ π = ((DVecHβπΎ)βπ) |
dvhlvec.k | β’ (π β (πΎ β HL β§ π β π»)) |
Ref | Expression |
---|---|
dvhlvec | β’ (π β π β LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhlvec.k | . 2 β’ (π β (πΎ β HL β§ π β π»)) | |
2 | eqid 2738 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | dvhlvec.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | eqid 2738 | . . 3 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
5 | eqid 2738 | . . 3 β’ ((TEndoβπΎ)βπ) = ((TEndoβπΎ)βπ) | |
6 | dvhlvec.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
7 | eqid 2738 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
8 | eqid 2738 | . . 3 β’ (+gβ(Scalarβπ)) = (+gβ(Scalarβπ)) | |
9 | eqid 2738 | . . 3 β’ (+gβπ) = (+gβπ) | |
10 | eqid 2738 | . . 3 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
11 | eqid 2738 | . . 3 β’ (invgβ(Scalarβπ)) = (invgβ(Scalarβπ)) | |
12 | eqid 2738 | . . 3 β’ (.rβ(Scalarβπ)) = (.rβ(Scalarβπ)) | |
13 | eqid 2738 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | dvhlveclem 39467 | . 2 β’ ((πΎ β HL β§ π β π») β π β LVec) |
15 | 1, 14 | syl 17 | 1 β’ (π β π β LVec) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6492 Basecbs 17018 +gcplusg 17068 .rcmulr 17069 Scalarcsca 17071 Β·π cvsca 17072 0gc0g 17256 invgcminusg 18684 LVecclvec 20487 HLchlt 37708 LHypclh 38343 LTrncltrn 38460 TEndoctendo 39111 DVecHcdvh 39437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-riotaBAD 37311 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-undef 8172 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-n0 12348 df-z 12434 df-uz 12697 df-fz 13354 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-sca 17084 df-vsca 17085 df-0g 17258 df-proset 18119 df-poset 18137 df-plt 18154 df-lub 18170 df-glb 18171 df-join 18172 df-meet 18173 df-p0 18249 df-p1 18250 df-lat 18256 df-clat 18323 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-grp 18686 df-minusg 18687 df-mgp 19827 df-ur 19844 df-ring 19891 df-oppr 19973 df-dvdsr 19994 df-unit 19995 df-invr 20025 df-dvr 20036 df-drng 20111 df-lmod 20248 df-lvec 20488 df-oposet 37534 df-ol 37536 df-oml 37537 df-covers 37624 df-ats 37625 df-atl 37656 df-cvlat 37680 df-hlat 37709 df-llines 37857 df-lplanes 37858 df-lvols 37859 df-lines 37860 df-psubsp 37862 df-pmap 37863 df-padd 38155 df-lhyp 38347 df-laut 38348 df-ldil 38463 df-ltrn 38464 df-trl 38518 df-tendo 39114 df-edring 39116 df-dvech 39438 |
This theorem is referenced by: dvhlmod 39469 dih1dimatlem 39688 dihlspsnssN 39691 dihlspsnat 39692 dihpN 39695 dihlatat 39696 dochsat 39742 dochshpncl 39743 dochlkr 39744 dochkrshp 39745 dochkrshp3 39747 dvh2dimatN 39799 dvh3dim3N 39808 dochsatshp 39810 dochsatshpb 39811 dochexmidat 39818 dochexmidlem3 39821 dochsnkr 39831 dochsnkr2 39832 dochflcl 39834 dochfl1 39835 dochkr1 39837 dochkr1OLDN 39838 lcfl6lem 39857 lcfl7lem 39858 lcfl9a 39864 lclkrlem1 39865 lclkrlem2a 39866 lclkrlem2e 39870 lclkrlem2g 39872 lclkrlem2h 39873 lclkrlem2o 39880 lclkrlem2p 39881 lclkrlem2q 39882 lclkrlem2s 39884 lclkrlem2v 39887 lclkrslem1 39896 lcfrvalsnN 39900 lcfrlem16 39917 lcfrlem20 39921 lcfrlem25 39926 lcfrlem29 39930 lcfrlem31 39932 lcfrlem33 39934 lcfrlem35 39936 lcdlvec 39950 lcdlkreqN 39981 lcdlkreq2N 39982 mapdordlem2 39996 mapdsn3 40002 mapdrvallem2 40004 mapdcnvatN 40025 mapdat 40026 mapdpglem10 40040 mapdpglem15 40045 mapdpglem17N 40047 mapdpglem18 40048 mapdpglem19 40049 mapdpglem21 40051 mapdpglem22 40052 mapdheq4lem 40090 mapdheq4 40091 mapdh6lem1N 40092 mapdh6lem2N 40093 mapdh6aN 40094 mapdh6b0N 40095 mapdh6bN 40096 mapdh6cN 40097 mapdh6dN 40098 mapdh6eN 40099 mapdh6fN 40100 mapdh6hN 40102 mapdh7eN 40107 mapdh7dN 40109 mapdh7fN 40110 mapdh75fN 40114 mapdh8aa 40135 mapdh8ab 40136 mapdh8ad 40138 mapdh8b 40139 mapdh8c 40140 mapdh8d0N 40141 mapdh8d 40142 mapdh8e 40143 mapdh9a 40148 mapdh9aOLDN 40149 hdmap1eq4N 40165 hdmap1l6lem1 40166 hdmap1l6lem2 40167 hdmap1l6a 40168 hdmap1l6b0N 40169 hdmap1l6b 40170 hdmap1l6c 40171 hdmap1l6d 40172 hdmap1l6e 40173 hdmap1l6f 40174 hdmap1l6h 40176 hdmap1eulemOLDN 40182 hdmapval0 40192 hdmapval3lemN 40196 hdmap10lem 40198 hdmap11lem1 40200 hdmap11lem2 40201 hdmaprnlem4N 40212 hdmaprnlem3eN 40217 hdmap14lem1a 40225 hdmap14lem4a 40230 hdmap14lem11 40237 hgmap11 40261 hdmaplkr 40272 hdmapip1 40275 hgmapvvlem1 40282 hgmapvvlem2 40283 hgmapvvlem3 40284 hlhillvec 40314 |
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