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Theorem hdmapinvlem4 31393
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 31308. 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 2284 . . . 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 30579 . . . . 5  |-  ( ph  ->  U  e.  LMod )
10 hdmapinvlem3.j . . . . 5  |-  ( ph  ->  J  e.  B )
11 eqid 2284 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2284 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
13 eqid 2284 . . . . . . 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 30581 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
16 eldifi 3299 . . . . . 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 15640 . . . . 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 3761 . . . . . 6  |-  ( ph  ->  { E }  C_  V )
23 hdmapinvlem3.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
241, 2, 3, 23dochssv 30824 . . . . . 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 3182 . . . 4  |-  ( ph  ->  D  e.  V )
28 hdmapinvlem3.i . . . . . 6  |-  ( ph  ->  I  e.  B )
293, 5, 18, 19lmodvscl 15640 . . . . . 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 3182 . . . . 5  |-  ( ph  ->  C  e.  V )
33 hdmapinvlem3.p . . . . . 6  |-  .+  =  ( +g  `  U )
343, 33lmodvacl 15637 . . . . 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 31380 . . 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 31392 . . . 4  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )
423, 4lmodvsubcl 15666 . . . . . 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 31389 . . . 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 31381 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
) ) )
47 eqid 2284 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
481, 2, 3, 33, 5, 47, 7, 8, 17, 30, 32hdmaplna2 31382 . . . . . . 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 31391 . . . . . . . 8  |-  ( ph  ->  ( ( S `  C ) `  E
)  =  .0.  )
5049oveq2d 5836 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R ) ( ( S `  C ) `
 E ) )  =  ( ( ( S `  ( I 
.x.  E ) ) `
 E ) ( +g  `  R )  .0.  ) )
515lmodrng 15631 . . . . . . . . . . 11  |-  ( U  e.  LMod  ->  R  e. 
Ring )
529, 51syl 15 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
53 rnggrp 15342 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
5452, 53syl 15 . . . . . . . . 9  |-  ( ph  ->  R  e.  Grp )
551, 2, 3, 5, 19, 7, 8, 17, 30hdmapipcl 31377 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )
5619, 47, 38grprid 14509 . . . . . . . . 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 31383 . . . . . . . 8  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  =  ( ( ( S `  E ) `
 E )  .X.  ( G `  I ) ) )
59 eqid 2284 . . . . . . . . . . 11  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
60 eqid 2284 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
611, 14, 59, 7, 8, 2, 5, 60hdmapevec2 31308 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  E ) `  E
)  =  ( 1r
`  R ) )
6261oveq1d 5835 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( ( 1r `  R ) 
.X.  ( G `  I ) ) )
631, 2, 5, 19, 39, 8, 28hgmapcl 31361 . . . . . . . . . 10  |-  ( ph  ->  ( G `  I
)  e.  B )
6419, 37, 60rnglidm 15360 . . . . . . . . . 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 2316 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6757, 58, 663eqtrd 2320 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( G `  I ) )
6848, 50, 673eqtrd 2320 . . . . . 6  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( G `
 I ) )
6968oveq2d 5836 . . . . 5  |-  ( ph  ->  ( J  .X.  (
( S `  (
( I  .x.  E
)  .+  C )
) `  E )
)  =  ( J 
.X.  ( G `  I ) ) )
7046, 69eqtrd 2316 . . . 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 31382 . . . . 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 31383 . . . . . . 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 31390 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  D
)  =  .0.  )
7473oveq1d 5835 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  D )  .X.  ( G `  I )
)  =  (  .0.  .X.  ( G `  I
) ) )
7519, 37, 38rnglz 15373 . . . . . . . 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 2320 . . . . . 6  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  .0.  )
7877oveq1d 5835 . . . . 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 31377 . . . . . 6  |-  ( ph  ->  ( ( S `  C ) `  D
)  e.  B )
8019, 47, 38grplid 14508 . . . . . 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 2320 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( S `  C ) `
 D ) )
8370, 82oveq12d 5838 . . 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 2325 . 2  |-  ( ph  ->  ( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) )  =  .0.  )
855, 19, 37lmodmcl 15635 . . . 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 14548 . . 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 1685    \ cdif 3150    C_ wss 3153   {csn 3641   <.cop 3644    _I cid 4303    |` cres 4690   ` cfv 5221  (class class class)co 5820   Basecbs 13144   +g cplusg 13204   .rcmulr 13205  Scalarcsca 13207   .scvsca 13208   0gc0g 13396   Grpcgrp 14358   -gcsg 14361   Ringcrg 15333   1rcur 15335   LModclmod 15623   HLchlt 28819   LHypclh 29452   LTrncltrn 29569   DVecHcdvh 30547   ocHcoch 30816  HVMapchvm 31225  HDMapchdma 31262  HGMapchg 31355
This theorem is referenced by:  hdmapglem5  31394  hgmapvvlem1  31395
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-ot 3651  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-undef 6292  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10779  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-sca 13220  df-vsca 13221  df-0g 13400  df-mre 13484  df-mrc 13485  df-acs 13487  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-mnd 14363  df-submnd 14412  df-grp 14485  df-minusg 14486  df-sbg 14487  df-subg 14614  df-cntz 14789  df-oppg 14815  df-lsm 14943  df-cmn 15087  df-abl 15088  df-mgp 15322  df-rng 15336  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-dvr 15461  df-drng 15510  df-lmod 15625  df-lss 15686  df-lsp 15725  df-lvec 15852  df-lsatoms 28445  df-lshyp 28446  df-lcv 28488  df-lfl 28527  df-lkr 28555  df-ldual 28593  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-llines 28966  df-lplanes 28967  df-lvols 28968  df-lines 28969  df-psubsp 28971  df-pmap 28972  df-padd 29264  df-lhyp 29456  df-laut 29457  df-ldil 29572  df-ltrn 29573  df-trl 29627  df-tgrp 30211  df-tendo 30223  df-edring 30225  df-dveca 30471  df-disoa 30498  df-dvech 30548  df-dib 30608  df-dic 30642  df-dih 30698  df-doch 30817  df-djh 30864  df-lcdual 31056  df-mapd 31094  df-hvmap 31226  df-hdmap1 31263  df-hdmap 31264  df-hgmap 31356
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