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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 39457 | . 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 20486 HLchlt 37698 LHypclh 38333 LTrncltrn 38450 TEndoctendo 39101 DVecHcdvh 39427 |
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 37301 |
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 19826 df-ur 19843 df-ring 19890 df-oppr 19972 df-dvdsr 19993 df-unit 19994 df-invr 20024 df-dvr 20035 df-drng 20110 df-lmod 20247 df-lvec 20487 df-oposet 37524 df-ol 37526 df-oml 37527 df-covers 37614 df-ats 37615 df-atl 37646 df-cvlat 37670 df-hlat 37699 df-llines 37847 df-lplanes 37848 df-lvols 37849 df-lines 37850 df-psubsp 37852 df-pmap 37853 df-padd 38145 df-lhyp 38337 df-laut 38338 df-ldil 38453 df-ltrn 38454 df-trl 38508 df-tendo 39104 df-edring 39106 df-dvech 39428 |
This theorem is referenced by: dvhlmod 39459 dih1dimatlem 39678 dihlspsnssN 39681 dihlspsnat 39682 dihpN 39685 dihlatat 39686 dochsat 39732 dochshpncl 39733 dochlkr 39734 dochkrshp 39735 dochkrshp3 39737 dvh2dimatN 39789 dvh3dim3N 39798 dochsatshp 39800 dochsatshpb 39801 dochexmidat 39808 dochexmidlem3 39811 dochsnkr 39821 dochsnkr2 39822 dochflcl 39824 dochfl1 39825 dochkr1 39827 dochkr1OLDN 39828 lcfl6lem 39847 lcfl7lem 39848 lcfl9a 39854 lclkrlem1 39855 lclkrlem2a 39856 lclkrlem2e 39860 lclkrlem2g 39862 lclkrlem2h 39863 lclkrlem2o 39870 lclkrlem2p 39871 lclkrlem2q 39872 lclkrlem2s 39874 lclkrlem2v 39877 lclkrslem1 39886 lcfrvalsnN 39890 lcfrlem16 39907 lcfrlem20 39911 lcfrlem25 39916 lcfrlem29 39920 lcfrlem31 39922 lcfrlem33 39924 lcfrlem35 39926 lcdlvec 39940 lcdlkreqN 39971 lcdlkreq2N 39972 mapdordlem2 39986 mapdsn3 39992 mapdrvallem2 39994 mapdcnvatN 40015 mapdat 40016 mapdpglem10 40030 mapdpglem15 40035 mapdpglem17N 40037 mapdpglem18 40038 mapdpglem19 40039 mapdpglem21 40041 mapdpglem22 40042 mapdheq4lem 40080 mapdheq4 40081 mapdh6lem1N 40082 mapdh6lem2N 40083 mapdh6aN 40084 mapdh6b0N 40085 mapdh6bN 40086 mapdh6cN 40087 mapdh6dN 40088 mapdh6eN 40089 mapdh6fN 40090 mapdh6hN 40092 mapdh7eN 40097 mapdh7dN 40099 mapdh7fN 40100 mapdh75fN 40104 mapdh8aa 40125 mapdh8ab 40126 mapdh8ad 40128 mapdh8b 40129 mapdh8c 40130 mapdh8d0N 40131 mapdh8d 40132 mapdh8e 40133 mapdh9a 40138 mapdh9aOLDN 40139 hdmap1eq4N 40155 hdmap1l6lem1 40156 hdmap1l6lem2 40157 hdmap1l6a 40158 hdmap1l6b0N 40159 hdmap1l6b 40160 hdmap1l6c 40161 hdmap1l6d 40162 hdmap1l6e 40163 hdmap1l6f 40164 hdmap1l6h 40166 hdmap1eulemOLDN 40172 hdmapval0 40182 hdmapval3lemN 40186 hdmap10lem 40188 hdmap11lem1 40190 hdmap11lem2 40191 hdmaprnlem4N 40202 hdmaprnlem3eN 40207 hdmap14lem1a 40215 hdmap14lem4a 40220 hdmap14lem11 40227 hgmap11 40251 hdmaplkr 40262 hdmapip1 40265 hgmapvvlem1 40272 hgmapvvlem2 40273 hgmapvvlem3 40274 hlhillvec 40304 |
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