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Theorem hdmapinvlem4 32039
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 31954. 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 2387 . . . 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 31225 . . . . 5  |-  ( ph  ->  U  e.  LMod )
10 hdmapinvlem3.j . . . . 5  |-  ( ph  ->  J  e.  B )
11 eqid 2387 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2387 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
13 eqid 2387 . . . . . . 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 31227 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
1615eldifad 3275 . . . . 5  |-  ( ph  ->  E  e.  V )
17 hdmapinvlem3.q . . . . . 6  |-  .x.  =  ( .s `  U )
18 hdmapinvlem3.b . . . . . 6  |-  B  =  ( Base `  R
)
193, 5, 17, 18lmodvscl 15894 . . . . 5  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  E  e.  V )  ->  ( J  .x.  E )  e.  V )
209, 10, 16, 19syl3anc 1184 . . . 4  |-  ( ph  ->  ( J  .x.  E
)  e.  V )
2116snssd 3886 . . . . . 6  |-  ( ph  ->  { E }  C_  V )
22 hdmapinvlem3.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
231, 2, 3, 22dochssv 31470 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  C_  V
)
248, 21, 23syl2anc 643 . . . . 5  |-  ( ph  ->  ( O `  { E } )  C_  V
)
25 hdmapinvlem3.d . . . . 5  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
2624, 25sseldd 3292 . . . 4  |-  ( ph  ->  D  e.  V )
27 hdmapinvlem3.i . . . . . 6  |-  ( ph  ->  I  e.  B )
283, 5, 17, 18lmodvscl 15894 . . . . . 6  |-  ( ( U  e.  LMod  /\  I  e.  B  /\  E  e.  V )  ->  (
I  .x.  E )  e.  V )
299, 27, 16, 28syl3anc 1184 . . . . 5  |-  ( ph  ->  ( I  .x.  E
)  e.  V )
30 hdmapinvlem3.c . . . . . 6  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
3124, 30sseldd 3292 . . . . 5  |-  ( ph  ->  C  e.  V )
32 hdmapinvlem3.p . . . . . 6  |-  .+  =  ( +g  `  U )
333, 32lmodvacl 15891 . . . . 5  |-  ( ( U  e.  LMod  /\  (
I  .x.  E )  e.  V  /\  C  e.  V )  ->  (
( I  .x.  E
)  .+  C )  e.  V )
349, 29, 31, 33syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( I  .x.  E )  .+  C
)  e.  V )
351, 2, 3, 4, 5, 6, 7, 8, 20, 26, 34hdmaplns1 32026 . . 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
) ) )
36 hdmapinvlem3.t . . . . 5  |-  .X.  =  ( .r `  R )
37 hdmapinvlem3.z . . . . 5  |-  .0.  =  ( 0g `  R )
38 hdmapinvlem3.g . . . . 5  |-  G  =  ( (HGMap `  K
) `  W )
39 hdmapinvlem3.ij . . . . 5  |-  ( ph  ->  ( I  .X.  ( G `  J )
)  =  ( ( S `  D ) `
 C ) )
401, 14, 22, 2, 3, 32, 4, 17, 5, 18, 36, 37, 7, 38, 8, 30, 25, 27, 10, 39hdmapinvlem3 32038 . . . 4  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )
413, 4lmodvsubcl 15916 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( J  .x.  E )  e.  V  /\  D  e.  V )  ->  (
( J  .x.  E
)  .-  D )  e.  V )
429, 20, 26, 41syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( J  .x.  E )  .-  D
)  e.  V )
431, 2, 3, 5, 37, 7, 8, 42, 34hdmapip0com 32035 . . . 4  |-  ( ph  ->  ( ( ( S `
 ( ( J 
.x.  E )  .-  D ) ) `  ( ( I  .x.  E )  .+  C
) )  =  .0.  <->  ( ( S `  (
( I  .x.  E
)  .+  C )
) `  ( ( J  .x.  E )  .-  D ) )  =  .0.  ) )
4440, 43mpbid 202 . . 3  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  (
( J  .x.  E
)  .-  D )
)  =  .0.  )
451, 2, 3, 17, 5, 18, 36, 7, 8, 16, 34, 10hdmaplnm1 32027 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
) ) )
46 eqid 2387 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
471, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31hdmaplna2 32028 . . . . . . 7  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( ( ( S `  (
I  .x.  E )
) `  E )
( +g  `  R ) ( ( S `  C ) `  E
) ) )
481, 14, 22, 2, 3, 5, 18, 36, 37, 7, 8, 30hdmapinvlem2 32037 . . . . . . . 8  |-  ( ph  ->  ( ( S `  C ) `  E
)  =  .0.  )
4948oveq2d 6036 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R ) ( ( S `  C ) `
 E ) )  =  ( ( ( S `  ( I 
.x.  E ) ) `
 E ) ( +g  `  R )  .0.  ) )
505lmodrng 15885 . . . . . . . . . . 11  |-  ( U  e.  LMod  ->  R  e. 
Ring )
519, 50syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
52 rnggrp 15596 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
5351, 52syl 16 . . . . . . . . 9  |-  ( ph  ->  R  e.  Grp )
541, 2, 3, 5, 18, 7, 8, 16, 29hdmapipcl 32023 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )
5518, 46, 37grprid 14763 . . . . . . . . 9  |-  ( ( R  e.  Grp  /\  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )  -> 
( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( ( S `
 ( I  .x.  E ) ) `  E ) )
5653, 54, 55syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( ( S `
 ( I  .x.  E ) ) `  E ) )
571, 2, 3, 17, 5, 18, 36, 7, 38, 8, 16, 16, 27hdmapglnm2 32029 . . . . . . . 8  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  =  ( ( ( S `  E ) `
 E )  .X.  ( G `  I ) ) )
58 eqid 2387 . . . . . . . . . . 11  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
59 eqid 2387 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
601, 14, 58, 7, 8, 2, 5, 59hdmapevec2 31954 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  E ) `  E
)  =  ( 1r
`  R ) )
6160oveq1d 6035 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( ( 1r `  R ) 
.X.  ( G `  I ) ) )
621, 2, 5, 18, 38, 8, 27hgmapcl 32007 . . . . . . . . . 10  |-  ( ph  ->  ( G `  I
)  e.  B )
6318, 36, 59rnglidm 15614 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  B )  ->  (
( 1r `  R
)  .X.  ( G `  I ) )  =  ( G `  I
) )
6451, 62, 63syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6561, 64eqtrd 2419 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6656, 57, 653eqtrd 2423 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( G `  I ) )
6747, 49, 663eqtrd 2423 . . . . . 6  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( G `
 I ) )
6867oveq2d 6036 . . . . 5  |-  ( ph  ->  ( J  .X.  (
( S `  (
( I  .x.  E
)  .+  C )
) `  E )
)  =  ( J 
.X.  ( G `  I ) ) )
6945, 68eqtrd 2419 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( G `  I ) ) )
701, 2, 3, 32, 5, 46, 7, 8, 26, 29, 31hdmaplna2 32028 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( ( S `  (
I  .x.  E )
) `  D )
( +g  `  R ) ( ( S `  C ) `  D
) ) )
711, 2, 3, 17, 5, 18, 36, 7, 38, 8, 26, 16, 27hdmapglnm2 32029 . . . . . . 7  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  ( ( ( S `  E ) `
 D )  .X.  ( G `  I ) ) )
721, 14, 22, 2, 3, 5, 18, 36, 37, 7, 8, 25hdmapinvlem1 32036 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  D
)  =  .0.  )
7372oveq1d 6035 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  D )  .X.  ( G `  I )
)  =  (  .0.  .X.  ( G `  I
) ) )
7418, 36, 37rnglz 15627 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  B )  ->  (  .0.  .X.  ( G `  I ) )  =  .0.  )
7551, 62, 74syl2anc 643 . . . . . . 7  |-  ( ph  ->  (  .0.  .X.  ( G `  I )
)  =  .0.  )
7671, 73, 753eqtrd 2423 . . . . . 6  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  .0.  )
7776oveq1d 6035 . . . . 5  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  D ) ( +g  `  R ) ( ( S `  C ) `
 D ) )  =  (  .0.  ( +g  `  R ) ( ( S `  C
) `  D )
) )
781, 2, 3, 5, 18, 7, 8, 26, 31hdmapipcl 32023 . . . . . 6  |-  ( ph  ->  ( ( S `  C ) `  D
)  e.  B )
7918, 46, 37grplid 14762 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( S `  C ) `  D
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( S `  C
) `  D )
)  =  ( ( S `  C ) `
 D ) )
8053, 78, 79syl2anc 643 . . . . 5  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( ( S `  C ) `
 D ) )  =  ( ( S `
 C ) `  D ) )
8170, 77, 803eqtrd 2423 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( S `  C ) `
 D ) )
8269, 81oveq12d 6038 . . 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 ) ) )
8335, 44, 823eqtr3rd 2428 . 2  |-  ( ph  ->  ( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) )  =  .0.  )
845, 18, 36lmodmcl 15889 . . . 4  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  ( G `  I )  e.  B )  ->  ( J  .X.  ( G `  I ) )  e.  B )
859, 10, 62, 84syl3anc 1184 . . 3  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  e.  B )
8618, 37, 6grpsubeq0 14802 . . 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 ) ) )
8753, 85, 78, 86syl3anc 1184 . 2  |-  ( ph  ->  ( ( ( J 
.X.  ( G `  I ) ) (
-g `  R )
( ( S `  C ) `  D
) )  =  .0.  <->  ( J  .X.  ( G `  I ) )  =  ( ( S `  C ) `  D
) ) )
8883, 87mpbid 202 1  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  =  ( ( S `  C ) `
 D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   {csn 3757   <.cop 3760    _I cid 4434    |` cres 4820   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   .rcmulr 13457  Scalarcsca 13459   .scvsca 13460   0gc0g 13650   Grpcgrp 14612   -gcsg 14615   Ringcrg 15587   1rcur 15589   LModclmod 15877   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   DVecHcdvh 31193   ocHcoch 31462  HVMapchvm 31871  HDMapchdma 31908  HGMapchg 32001
This theorem is referenced by:  hdmapglem5  32040  hgmapvvlem1  32041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-ot 3767  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-tpos 6415  df-undef 6479  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-0g 13654  df-mre 13738  df-mrc 13739  df-acs 13741  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-cntz 15043  df-oppg 15069  df-lsm 15197  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-dvr 15715  df-drng 15764  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lvec 16102  df-lsatoms 29091  df-lshyp 29092  df-lcv 29134  df-lfl 29173  df-lkr 29201  df-ldual 29239  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tgrp 30857  df-tendo 30869  df-edring 30871  df-dveca 31117  df-disoa 31144  df-dvech 31194  df-dib 31254  df-dic 31288  df-dih 31344  df-doch 31463  df-djh 31510  df-lcdual 31702  df-mapd 31740  df-hvmap 31872  df-hdmap1 31909  df-hdmap 31910  df-hgmap 32002
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