Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmapinvlem4 Unicode version

Theorem hdmapinvlem4 31487
Description: Part 1.1 of Proposition 1 of [Baer] p. 110. We use  C,  D,  I, and  J for Baer's u, v, s, and t. Our unit vector  E has the required properties for his w by hdmapevec2 31402. Our  ( ( S `  D ) `  C ) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)
Hypotheses
Ref Expression
hdmapinvlem3.h  |-  H  =  ( LHyp `  K
)
hdmapinvlem3.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmapinvlem3.o  |-  O  =  ( ( ocH `  K
) `  W )
hdmapinvlem3.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmapinvlem3.v  |-  V  =  ( Base `  U
)
hdmapinvlem3.p  |-  .+  =  ( +g  `  U )
hdmapinvlem3.m  |-  .-  =  ( -g `  U )
hdmapinvlem3.q  |-  .x.  =  ( .s `  U )
hdmapinvlem3.r  |-  R  =  (Scalar `  U )
hdmapinvlem3.b  |-  B  =  ( Base `  R
)
hdmapinvlem3.t  |-  .X.  =  ( .r `  R )
hdmapinvlem3.z  |-  .0.  =  ( 0g `  R )
hdmapinvlem3.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmapinvlem3.g  |-  G  =  ( (HGMap `  K
) `  W )
hdmapinvlem3.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmapinvlem3.c  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
hdmapinvlem3.d  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
hdmapinvlem3.i  |-  ( ph  ->  I  e.  B )
hdmapinvlem3.j  |-  ( ph  ->  J  e.  B )
hdmapinvlem3.ij  |-  ( ph  ->  ( I  .X.  ( G `  J )
)  =  ( ( S `  D ) `
 C ) )
Assertion
Ref Expression
hdmapinvlem4  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  =  ( ( S `  C ) `
 D ) )

Proof of Theorem hdmapinvlem4
StepHypRef Expression
1 hdmapinvlem3.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmapinvlem3.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmapinvlem3.v . . . 4  |-  V  =  ( Base `  U
)
4 hdmapinvlem3.m . . . 4  |-  .-  =  ( -g `  U )
5 hdmapinvlem3.r . . . 4  |-  R  =  (Scalar `  U )
6 eqid 2283 . . . 4  |-  ( -g `  R )  =  (
-g `  R )
7 hdmapinvlem3.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
8 hdmapinvlem3.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
91, 2, 8dvhlmod 30673 . . . . 5  |-  ( ph  ->  U  e.  LMod )
10 hdmapinvlem3.j . . . . 5  |-  ( ph  ->  J  e.  B )
11 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2283 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
13 eqid 2283 . . . . . . 7  |-  ( 0g
`  U )  =  ( 0g `  U
)
14 hdmapinvlem3.e . . . . . . 7  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
151, 11, 12, 2, 3, 13, 14, 8dvheveccl 30675 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
16 eldifi 3298 . . . . . 6  |-  ( E  e.  ( V  \  { ( 0g `  U ) } )  ->  E  e.  V
)
1715, 16syl 15 . . . . 5  |-  ( ph  ->  E  e.  V )
18 hdmapinvlem3.q . . . . . 6  |-  .x.  =  ( .s `  U )
19 hdmapinvlem3.b . . . . . 6  |-  B  =  ( Base `  R
)
203, 5, 18, 19lmodvscl 15644 . . . . 5  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  E  e.  V )  ->  ( J  .x.  E )  e.  V )
219, 10, 17, 20syl3anc 1182 . . . 4  |-  ( ph  ->  ( J  .x.  E
)  e.  V )
2217snssd 3760 . . . . . 6  |-  ( ph  ->  { E }  C_  V )
23 hdmapinvlem3.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
241, 2, 3, 23dochssv 30918 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  C_  V
)
258, 22, 24syl2anc 642 . . . . 5  |-  ( ph  ->  ( O `  { E } )  C_  V
)
26 hdmapinvlem3.d . . . . 5  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
2725, 26sseldd 3181 . . . 4  |-  ( ph  ->  D  e.  V )
28 hdmapinvlem3.i . . . . . 6  |-  ( ph  ->  I  e.  B )
293, 5, 18, 19lmodvscl 15644 . . . . . 6  |-  ( ( U  e.  LMod  /\  I  e.  B  /\  E  e.  V )  ->  (
I  .x.  E )  e.  V )
309, 28, 17, 29syl3anc 1182 . . . . 5  |-  ( ph  ->  ( I  .x.  E
)  e.  V )
31 hdmapinvlem3.c . . . . . 6  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
3225, 31sseldd 3181 . . . . 5  |-  ( ph  ->  C  e.  V )
33 hdmapinvlem3.p . . . . . 6  |-  .+  =  ( +g  `  U )
343, 33lmodvacl 15641 . . . . 5  |-  ( ( U  e.  LMod  /\  (
I  .x.  E )  e.  V  /\  C  e.  V )  ->  (
( I  .x.  E
)  .+  C )  e.  V )
359, 30, 32, 34syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( I  .x.  E )  .+  C
)  e.  V )
361, 2, 3, 4, 5, 6, 7, 8, 21, 27, 35hdmaplns1 31474 . . 3  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  (
( J  .x.  E
)  .-  D )
)  =  ( ( ( S `  (
( I  .x.  E
)  .+  C )
) `  ( J  .x.  E ) ) (
-g `  R )
( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
) ) )
37 hdmapinvlem3.t . . . . 5  |-  .X.  =  ( .r `  R )
38 hdmapinvlem3.z . . . . 5  |-  .0.  =  ( 0g `  R )
39 hdmapinvlem3.g . . . . 5  |-  G  =  ( (HGMap `  K
) `  W )
40 hdmapinvlem3.ij . . . . 5  |-  ( ph  ->  ( I  .X.  ( G `  J )
)  =  ( ( S `  D ) `
 C ) )
411, 14, 23, 2, 3, 33, 4, 18, 5, 19, 37, 38, 7, 39, 8, 31, 26, 28, 10, 40hdmapinvlem3 31486 . . . 4  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )
423, 4lmodvsubcl 15670 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( J  .x.  E )  e.  V  /\  D  e.  V )  ->  (
( J  .x.  E
)  .-  D )  e.  V )
439, 21, 27, 42syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( J  .x.  E )  .-  D
)  e.  V )
441, 2, 3, 5, 38, 7, 8, 43, 35hdmapip0com 31483 . . . 4  |-  ( ph  ->  ( ( ( S `
 ( ( J 
.x.  E )  .-  D ) ) `  ( ( I  .x.  E )  .+  C
) )  =  .0.  <->  ( ( S `  (
( I  .x.  E
)  .+  C )
) `  ( ( J  .x.  E )  .-  D ) )  =  .0.  ) )
4541, 44mpbid 201 . . 3  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  (
( J  .x.  E
)  .-  D )
)  =  .0.  )
461, 2, 3, 18, 5, 19, 37, 7, 8, 17, 35, 10hdmaplnm1 31475 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
) ) )
47 eqid 2283 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
481, 2, 3, 33, 5, 47, 7, 8, 17, 30, 32hdmaplna2 31476 . . . . . . 7  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( ( ( S `  (
I  .x.  E )
) `  E )
( +g  `  R ) ( ( S `  C ) `  E
) ) )
491, 14, 23, 2, 3, 5, 19, 37, 38, 7, 8, 31hdmapinvlem2 31485 . . . . . . . 8  |-  ( ph  ->  ( ( S `  C ) `  E
)  =  .0.  )
5049oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R ) ( ( S `  C ) `
 E ) )  =  ( ( ( S `  ( I 
.x.  E ) ) `
 E ) ( +g  `  R )  .0.  ) )
515lmodrng 15635 . . . . . . . . . . 11  |-  ( U  e.  LMod  ->  R  e. 
Ring )
529, 51syl 15 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
53 rnggrp 15346 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
5452, 53syl 15 . . . . . . . . 9  |-  ( ph  ->  R  e.  Grp )
551, 2, 3, 5, 19, 7, 8, 17, 30hdmapipcl 31471 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )
5619, 47, 38grprid 14513 . . . . . . . . 9  |-  ( ( R  e.  Grp  /\  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )  -> 
( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( ( S `
 ( I  .x.  E ) ) `  E ) )
5754, 55, 56syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( ( S `
 ( I  .x.  E ) ) `  E ) )
581, 2, 3, 18, 5, 19, 37, 7, 39, 8, 17, 17, 28hdmapglnm2 31477 . . . . . . . 8  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  =  ( ( ( S `  E ) `
 E )  .X.  ( G `  I ) ) )
59 eqid 2283 . . . . . . . . . . 11  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
60 eqid 2283 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
611, 14, 59, 7, 8, 2, 5, 60hdmapevec2 31402 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  E ) `  E
)  =  ( 1r
`  R ) )
6261oveq1d 5873 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( ( 1r `  R ) 
.X.  ( G `  I ) ) )
631, 2, 5, 19, 39, 8, 28hgmapcl 31455 . . . . . . . . . 10  |-  ( ph  ->  ( G `  I
)  e.  B )
6419, 37, 60rnglidm 15364 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  B )  ->  (
( 1r `  R
)  .X.  ( G `  I ) )  =  ( G `  I
) )
6552, 63, 64syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6662, 65eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6757, 58, 663eqtrd 2319 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( G `  I ) )
6848, 50, 673eqtrd 2319 . . . . . 6  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( G `
 I ) )
6968oveq2d 5874 . . . . 5  |-  ( ph  ->  ( J  .X.  (
( S `  (
( I  .x.  E
)  .+  C )
) `  E )
)  =  ( J 
.X.  ( G `  I ) ) )
7046, 69eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( G `  I ) ) )
711, 2, 3, 33, 5, 47, 7, 8, 27, 30, 32hdmaplna2 31476 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( ( S `  (
I  .x.  E )
) `  D )
( +g  `  R ) ( ( S `  C ) `  D
) ) )
721, 2, 3, 18, 5, 19, 37, 7, 39, 8, 27, 17, 28hdmapglnm2 31477 . . . . . . 7  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  ( ( ( S `  E ) `
 D )  .X.  ( G `  I ) ) )
731, 14, 23, 2, 3, 5, 19, 37, 38, 7, 8, 26hdmapinvlem1 31484 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  D
)  =  .0.  )
7473oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  D )  .X.  ( G `  I )
)  =  (  .0.  .X.  ( G `  I
) ) )
7519, 37, 38rnglz 15377 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  B )  ->  (  .0.  .X.  ( G `  I ) )  =  .0.  )
7652, 63, 75syl2anc 642 . . . . . . 7  |-  ( ph  ->  (  .0.  .X.  ( G `  I )
)  =  .0.  )
7772, 74, 763eqtrd 2319 . . . . . 6  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  .0.  )
7877oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  D ) ( +g  `  R ) ( ( S `  C ) `
 D ) )  =  (  .0.  ( +g  `  R ) ( ( S `  C
) `  D )
) )
791, 2, 3, 5, 19, 7, 8, 27, 32hdmapipcl 31471 . . . . . 6  |-  ( ph  ->  ( ( S `  C ) `  D
)  e.  B )
8019, 47, 38grplid 14512 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( S `  C ) `  D
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( S `  C
) `  D )
)  =  ( ( S `  C ) `
 D ) )
8154, 79, 80syl2anc 642 . . . . 5  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( ( S `  C ) `
 D ) )  =  ( ( S `
 C ) `  D ) )
8271, 78, 813eqtrd 2319 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( S `  C ) `
 D ) )
8370, 82oveq12d 5876 . . 3  |-  ( ph  ->  ( ( ( S `
 ( ( I 
.x.  E )  .+  C ) ) `  ( J  .x.  E ) ) ( -g `  R
) ( ( S `
 ( ( I 
.x.  E )  .+  C ) ) `  D ) )  =  ( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) ) )
8436, 45, 833eqtr3rd 2324 . 2  |-  ( ph  ->  ( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) )  =  .0.  )
855, 19, 37lmodmcl 15639 . . . 4  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  ( G `  I )  e.  B )  ->  ( J  .X.  ( G `  I ) )  e.  B )
869, 10, 63, 85syl3anc 1182 . . 3  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  e.  B )
8719, 38, 6grpsubeq0 14552 . . 3  |-  ( ( R  e.  Grp  /\  ( J  .X.  ( G `
 I ) )  e.  B  /\  (
( S `  C
) `  D )  e.  B )  ->  (
( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) )  =  .0.  <->  ( J  .X.  ( G `  I ) )  =  ( ( S `  C ) `
 D ) ) )
8854, 86, 79, 87syl3anc 1182 . 2  |-  ( ph  ->  ( ( ( J 
.X.  ( G `  I ) ) (
-g `  R )
( ( S `  C ) `  D
) )  =  .0.  <->  ( J  .X.  ( G `  I ) )  =  ( ( S `  C ) `  D
) ) )
8984, 88mpbid 201 1  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  =  ( ( S `  C ) `
 D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {csn 3640   <.cop 3643    _I cid 4304    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   -gcsg 14365   Ringcrg 15337   1rcur 15339   LModclmod 15627   HLchlt 28913   LHypclh 29546   LTrncltrn 29663   DVecHcdvh 30641   ocHcoch 30910  HVMapchvm 31319  HDMapchdma 31356  HGMapchg 31449
This theorem is referenced by:  hdmapglem5  31488  hgmapvvlem1  31489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 28539  df-lshyp 28540  df-lcv 28582  df-lfl 28621  df-lkr 28649  df-ldual 28687  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061  df-lvols 29062  df-lines 29063  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667  df-trl 29721  df-tgrp 30305  df-tendo 30317  df-edring 30319  df-dveca 30565  df-disoa 30592  df-dvech 30642  df-dib 30702  df-dic 30736  df-dih 30792  df-doch 30911  df-djh 30958  df-lcdual 31150  df-mapd 31188  df-hvmap 31320  df-hdmap1 31357  df-hdmap 31358  df-hgmap 31450
  Copyright terms: Public domain W3C validator