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Theorem hdmapinvlem4 32411
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 32326. 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 2408 . . . 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 31597 . . . . 5  |-  ( ph  ->  U  e.  LMod )
10 hdmapinvlem3.j . . . . 5  |-  ( ph  ->  J  e.  B )
11 eqid 2408 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2408 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
13 eqid 2408 . . . . . . 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 31599 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
1615eldifad 3296 . . . . 5  |-  ( ph  ->  E  e.  V )
17 hdmapinvlem3.q . . . . . 6  |-  .x.  =  ( .s `  U )
18 hdmapinvlem3.b . . . . . 6  |-  B  =  ( Base `  R
)
193, 5, 17, 18lmodvscl 15926 . . . . 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 3907 . . . . . 6  |-  ( ph  ->  { E }  C_  V )
22 hdmapinvlem3.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
231, 2, 3, 22dochssv 31842 . . . . . 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 3313 . . . 4  |-  ( ph  ->  D  e.  V )
27 hdmapinvlem3.i . . . . . 6  |-  ( ph  ->  I  e.  B )
283, 5, 17, 18lmodvscl 15926 . . . . . 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 3313 . . . . 5  |-  ( ph  ->  C  e.  V )
32 hdmapinvlem3.p . . . . . 6  |-  .+  =  ( +g  `  U )
333, 32lmodvacl 15923 . . . . 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 32398 . . 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 32410 . . . 4  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )
413, 4lmodvsubcl 15948 . . . . . 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 32407 . . . 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 32399 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
) ) )
46 eqid 2408 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
471, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31hdmaplna2 32400 . . . . . . 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 32409 . . . . . . . 8  |-  ( ph  ->  ( ( S `  C ) `  E
)  =  .0.  )
4948oveq2d 6060 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R ) ( ( S `  C ) `
 E ) )  =  ( ( ( S `  ( I 
.x.  E ) ) `
 E ) ( +g  `  R )  .0.  ) )
505lmodrng 15917 . . . . . . . . . . 11  |-  ( U  e.  LMod  ->  R  e. 
Ring )
519, 50syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
52 rnggrp 15628 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
5351, 52syl 16 . . . . . . . . 9  |-  ( ph  ->  R  e.  Grp )
541, 2, 3, 5, 18, 7, 8, 16, 29hdmapipcl 32395 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )
5518, 46, 37grprid 14795 . . . . . . . . 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 32401 . . . . . . . 8  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  =  ( ( ( S `  E ) `
 E )  .X.  ( G `  I ) ) )
58 eqid 2408 . . . . . . . . . . 11  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
59 eqid 2408 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
601, 14, 58, 7, 8, 2, 5, 59hdmapevec2 32326 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  E ) `  E
)  =  ( 1r
`  R ) )
6160oveq1d 6059 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( ( 1r `  R ) 
.X.  ( G `  I ) ) )
621, 2, 5, 18, 38, 8, 27hgmapcl 32379 . . . . . . . . . 10  |-  ( ph  ->  ( G `  I
)  e.  B )
6318, 36, 59rnglidm 15646 . . . . . . . . . 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 2440 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6656, 57, 653eqtrd 2444 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( G `  I ) )
6747, 49, 663eqtrd 2444 . . . . . 6  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( G `
 I ) )
6867oveq2d 6060 . . . . 5  |-  ( ph  ->  ( J  .X.  (
( S `  (
( I  .x.  E
)  .+  C )
) `  E )
)  =  ( J 
.X.  ( G `  I ) ) )
6945, 68eqtrd 2440 . . . 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 32400 . . . . 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 32401 . . . . . . 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 32408 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  D
)  =  .0.  )
7372oveq1d 6059 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  D )  .X.  ( G `  I )
)  =  (  .0.  .X.  ( G `  I
) ) )
7418, 36, 37rnglz 15659 . . . . . . . 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 2444 . . . . . 6  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  .0.  )
7776oveq1d 6059 . . . . 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 32395 . . . . . 6  |-  ( ph  ->  ( ( S `  C ) `  D
)  e.  B )
7918, 46, 37grplid 14794 . . . . . 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 2444 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( S `  C ) `
 D ) )
8269, 81oveq12d 6062 . . 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 2449 . 2  |-  ( ph  ->  ( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) )  =  .0.  )
845, 18, 36lmodmcl 15921 . . . 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 14834 . . 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 1721    C_ wss 3284   {csn 3778   <.cop 3781    _I cid 4457    |` cres 4843   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488   .rcmulr 13489  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   Grpcgrp 14644   -gcsg 14647   Ringcrg 15619   1rcur 15621   LModclmod 15909   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   DVecHcdvh 31565   ocHcoch 31834  HVMapchvm 32243  HDMapchdma 32280  HGMapchg 32373
This theorem is referenced by:  hdmapglem5  32412  hgmapvvlem1  32413
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-ot 3788  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lshyp 29464  df-lcv 29506  df-lfl 29545  df-lkr 29573  df-ldual 29611  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tgrp 31229  df-tendo 31241  df-edring 31243  df-dveca 31489  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716  df-doch 31835  df-djh 31882  df-lcdual 32074  df-mapd 32112  df-hvmap 32244  df-hdmap1 32281  df-hdmap 32282  df-hgmap 32374
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