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Theorem hdmapinvlem4 31244
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 31159. 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 2256 . . . 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 30430 . . . . 5  |-  ( ph  ->  U  e.  LMod )
10 hdmapinvlem3.j . . . . 5  |-  ( ph  ->  J  e.  B )
11 eqid 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2256 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
13 eqid 2256 . . . . . . 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 30432 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
16 eldifi 3240 . . . . . 6  |-  ( E  e.  ( V  \  { ( 0g `  U ) } )  ->  E  e.  V
)
1715, 16syl 17 . . . . 5  |-  ( ph  ->  E  e.  V )
18 hdmapinvlem3.q . . . . . 6  |-  .x.  =  ( .s `  U )
19 hdmapinvlem3.b . . . . . 6  |-  B  =  ( Base `  R
)
203, 5, 18, 19lmodvscl 15571 . . . . 5  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  E  e.  V )  ->  ( J  .x.  E )  e.  V )
219, 10, 17, 20syl3anc 1187 . . . 4  |-  ( ph  ->  ( J  .x.  E
)  e.  V )
2217snssd 3701 . . . . . 6  |-  ( ph  ->  { E }  C_  V )
23 hdmapinvlem3.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
241, 2, 3, 23dochssv 30675 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  C_  V
)
258, 22, 24syl2anc 645 . . . . 5  |-  ( ph  ->  ( O `  { E } )  C_  V
)
26 hdmapinvlem3.d . . . . 5  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
2725, 26sseldd 3123 . . . 4  |-  ( ph  ->  D  e.  V )
28 hdmapinvlem3.i . . . . . 6  |-  ( ph  ->  I  e.  B )
293, 5, 18, 19lmodvscl 15571 . . . . . 6  |-  ( ( U  e.  LMod  /\  I  e.  B  /\  E  e.  V )  ->  (
I  .x.  E )  e.  V )
309, 28, 17, 29syl3anc 1187 . . . . 5  |-  ( ph  ->  ( I  .x.  E
)  e.  V )
31 hdmapinvlem3.c . . . . . 6  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
3225, 31sseldd 3123 . . . . 5  |-  ( ph  ->  C  e.  V )
33 hdmapinvlem3.p . . . . . 6  |-  .+  =  ( +g  `  U )
343, 33lmodvacl 15568 . . . . 5  |-  ( ( U  e.  LMod  /\  (
I  .x.  E )  e.  V  /\  C  e.  V )  ->  (
( I  .x.  E
)  .+  C )  e.  V )
359, 30, 32, 34syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( I  .x.  E )  .+  C
)  e.  V )
361, 2, 3, 4, 5, 6, 7, 8, 21, 27, 35hdmaplns1 31231 . . 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 31243 . . . 4  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )
423, 4lmodvsubcl 15597 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( J  .x.  E )  e.  V  /\  D  e.  V )  ->  (
( J  .x.  E
)  .-  D )  e.  V )
439, 21, 27, 42syl3anc 1187 . . . . 5  |-  ( ph  ->  ( ( J  .x.  E )  .-  D
)  e.  V )
441, 2, 3, 5, 38, 7, 8, 43, 35hdmapip0com 31240 . . . 4  |-  ( ph  ->  ( ( ( S `
 ( ( J 
.x.  E )  .-  D ) ) `  ( ( I  .x.  E )  .+  C
) )  =  .0.  <->  ( ( S `  (
( I  .x.  E
)  .+  C )
) `  ( ( J  .x.  E )  .-  D ) )  =  .0.  ) )
4541, 44mpbid 203 . . 3  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  (
( J  .x.  E
)  .-  D )
)  =  .0.  )
461, 2, 3, 18, 5, 19, 37, 7, 8, 17, 35, 10hdmaplnm1 31232 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  ( J  .x.  E ) )  =  ( J  .X.  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
) ) )
47 eqid 2256 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
481, 2, 3, 33, 5, 47, 7, 8, 17, 30, 32hdmaplna2 31233 . . . . . . 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 31242 . . . . . . . 8  |-  ( ph  ->  ( ( S `  C ) `  E
)  =  .0.  )
5049oveq2d 5773 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R ) ( ( S `  C ) `
 E ) )  =  ( ( ( S `  ( I 
.x.  E ) ) `
 E ) ( +g  `  R )  .0.  ) )
515lmodrng 15562 . . . . . . . . . . 11  |-  ( U  e.  LMod  ->  R  e. 
Ring )
529, 51syl 17 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
53 rnggrp 15273 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  R  e. 
Grp )
5452, 53syl 17 . . . . . . . . 9  |-  ( ph  ->  R  e.  Grp )
551, 2, 3, 5, 19, 7, 8, 17, 30hdmapipcl 31228 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  e.  B )
5619, 47, 38grprid 14440 . . . . . . . . 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 645 . . . . . . . 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 31234 . . . . . . . 8  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  E )  =  ( ( ( S `  E ) `
 E )  .X.  ( G `  I ) ) )
59 eqid 2256 . . . . . . . . . . 11  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
60 eqid 2256 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
611, 14, 59, 7, 8, 2, 5, 60hdmapevec2 31159 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  E ) `  E
)  =  ( 1r
`  R ) )
6261oveq1d 5772 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( ( 1r `  R ) 
.X.  ( G `  I ) ) )
631, 2, 5, 19, 39, 8, 28hgmapcl 31212 . . . . . . . . . 10  |-  ( ph  ->  ( G `  I
)  e.  B )
6419, 37, 60rnglidm 15291 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  B )  ->  (
( 1r `  R
)  .X.  ( G `  I ) )  =  ( G `  I
) )
6552, 63, 64syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6662, 65eqtrd 2288 . . . . . . . 8  |-  ( ph  ->  ( ( ( S `
 E ) `  E )  .X.  ( G `  I )
)  =  ( G `
 I ) )
6757, 58, 663eqtrd 2292 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 ( I  .x.  E ) ) `  E ) ( +g  `  R )  .0.  )  =  ( G `  I ) )
6848, 50, 673eqtrd 2292 . . . . . 6  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  E
)  =  ( G `
 I ) )
6968oveq2d 5773 . . . . 5  |-  ( ph  ->  ( J  .X.  (
( S `  (
( I  .x.  E
)  .+  C )
) `  E )
)  =  ( J 
.X.  ( G `  I ) ) )
7046, 69eqtrd 2288 . . . 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 31233 . . . . 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 31234 . . . . . . 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 31241 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  D
)  =  .0.  )
7473oveq1d 5772 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  D )  .X.  ( G `  I )
)  =  (  .0.  .X.  ( G `  I
) ) )
7519, 37, 38rnglz 15304 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  B )  ->  (  .0.  .X.  ( G `  I ) )  =  .0.  )
7652, 63, 75syl2anc 645 . . . . . . 7  |-  ( ph  ->  (  .0.  .X.  ( G `  I )
)  =  .0.  )
7772, 74, 763eqtrd 2292 . . . . . 6  |-  ( ph  ->  ( ( S `  ( I  .x.  E ) ) `  D )  =  .0.  )
7877oveq1d 5772 . . . . 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 31228 . . . . . 6  |-  ( ph  ->  ( ( S `  C ) `  D
)  e.  B )
8019, 47, 38grplid 14439 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( S `  C ) `  D
)  e.  B )  ->  (  .0.  ( +g  `  R ) ( ( S `  C
) `  D )
)  =  ( ( S `  C ) `
 D ) )
8154, 79, 80syl2anc 645 . . . . 5  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( ( S `  C ) `
 D ) )  =  ( ( S `
 C ) `  D ) )
8271, 78, 813eqtrd 2292 . . . 4  |-  ( ph  ->  ( ( S `  ( ( I  .x.  E )  .+  C
) ) `  D
)  =  ( ( S `  C ) `
 D ) )
8370, 82oveq12d 5775 . . 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 2297 . 2  |-  ( ph  ->  ( ( J  .X.  ( G `  I ) ) ( -g `  R
) ( ( S `
 C ) `  D ) )  =  .0.  )
855, 19, 37lmodmcl 15566 . . . 4  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  ( G `  I )  e.  B )  ->  ( J  .X.  ( G `  I ) )  e.  B )
869, 10, 63, 85syl3anc 1187 . . 3  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  e.  B )
8719, 38, 6grpsubeq0 14479 . . 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 1187 . 2  |-  ( ph  ->  ( ( ( J 
.X.  ( G `  I ) ) (
-g `  R )
( ( S `  C ) `  D
) )  =  .0.  <->  ( J  .X.  ( G `  I ) )  =  ( ( S `  C ) `  D
) ) )
8984, 88mpbid 203 1  |-  ( ph  ->  ( J  .X.  ( G `  I )
)  =  ( ( S `  C ) `
 D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3091    C_ wss 3094   {csn 3581   <.cop 3584    _I cid 4241    |` cres 4628   ` cfv 4638  (class class class)co 5757   Basecbs 13075   +g cplusg 13135   .rcmulr 13136  Scalarcsca 13138   .scvsca 13139   0gc0g 13327   Grpcgrp 14289   -gcsg 14292   Ringcrg 15264   1rcur 15266   LModclmod 15554   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   DVecHcdvh 30398   ocHcoch 30667  HVMapchvm 31076  HDMapchdma 31113  HGMapchg 31206
This theorem is referenced by:  hdmapglem5  31245  hgmapvvlem1  31246
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-ot 3591  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-tpos 6133  df-iota 6190  df-undef 6229  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-n0 9898  df-z 9957  df-uz 10163  df-fz 10714  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-sca 13151  df-vsca 13152  df-0g 13331  df-mre 13415  df-mrc 13416  df-acs 13418  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-sbg 14418  df-subg 14545  df-cntz 14720  df-oppg 14746  df-lsm 14874  df-cmn 15018  df-abl 15019  df-mgp 15253  df-ring 15267  df-ur 15269  df-oppr 15332  df-dvdsr 15350  df-unit 15351  df-invr 15381  df-dvr 15392  df-drng 15441  df-lmod 15556  df-lss 15617  df-lsp 15656  df-lvec 15783  df-lsatoms 28296  df-lshyp 28297  df-lcv 28339  df-lfl 28378  df-lkr 28406  df-ldual 28444  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478  df-tgrp 30062  df-tendo 30074  df-edring 30076  df-dveca 30322  df-disoa 30349  df-dvech 30399  df-dib 30459  df-dic 30493  df-dih 30549  df-doch 30668  df-djh 30715  df-lcdual 30907  df-mapd 30945  df-hvmap 31077  df-hdmap1 31114  df-hdmap 31115  df-hgmap 31207
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