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Theorem hdmapinvlem3 30914
Description: Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (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
hdmapinvlem3  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )

Proof of Theorem hdmapinvlem3
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 eqid 2253 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
6 eqid 2253 . . . 4  |-  ( -g `  ( (LCDual `  K
) `  W )
)  =  ( -g `  ( (LCDual `  K
) `  W )
)
7 hdmapinvlem3.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
8 hdmapinvlem3.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
91, 2, 8dvhlmod 30101 . . . . 5  |-  ( ph  ->  U  e.  LMod )
10 hdmapinvlem3.j . . . . 5  |-  ( ph  ->  J  e.  B )
11 eqid 2253 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2253 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
13 eqid 2253 . . . . . . 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 30103 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
16 eldifi 3215 . . . . . 6  |-  ( E  e.  ( V  \  { ( 0g `  U ) } )  ->  E  e.  V
)
1715, 16syl 17 . . . . 5  |-  ( ph  ->  E  e.  V )
18 hdmapinvlem3.r . . . . . 6  |-  R  =  (Scalar `  U )
19 hdmapinvlem3.q . . . . . 6  |-  .x.  =  ( .s `  U )
20 hdmapinvlem3.b . . . . . 6  |-  B  =  ( Base `  R
)
213, 18, 19, 20lmodvscl 15479 . . . . 5  |-  ( ( U  e.  LMod  /\  J  e.  B  /\  E  e.  V )  ->  ( J  .x.  E )  e.  V )
229, 10, 17, 21syl3anc 1187 . . . 4  |-  ( ph  ->  ( J  .x.  E
)  e.  V )
2317snssd 3660 . . . . . 6  |-  ( ph  ->  { E }  C_  V )
24 hdmapinvlem3.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
251, 2, 3, 24dochssv 30346 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  C_  V
)
268, 23, 25syl2anc 645 . . . . 5  |-  ( ph  ->  ( O `  { E } )  C_  V
)
27 hdmapinvlem3.d . . . . 5  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
2826, 27sseldd 3104 . . . 4  |-  ( ph  ->  D  e.  V )
291, 2, 3, 4, 5, 6, 7, 8, 22, 28hdmapsub 30841 . . 3  |-  ( ph  ->  ( S `  (
( J  .x.  E
)  .-  D )
)  =  ( ( S `  ( J 
.x.  E ) ) ( -g `  (
(LCDual `  K ) `  W ) ) ( S `  D ) ) )
3029fveq1d 5379 . 2  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  ( ( ( S `  ( J  .x.  E ) ) ( -g `  (
(LCDual `  K ) `  W ) ) ( S `  D ) ) `  ( ( I  .x.  E ) 
.+  C ) ) )
31 eqid 2253 . . . 4  |-  ( -g `  R )  =  (
-g `  R )
32 eqid 2253 . . . 4  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
331, 2, 3, 5, 32, 7, 8, 22hdmapcl 30824 . . . 4  |-  ( ph  ->  ( S `  ( J  .x.  E ) )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
341, 2, 3, 5, 32, 7, 8, 28hdmapcl 30824 . . . 4  |-  ( ph  ->  ( S `  D
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
35 hdmapinvlem3.i . . . . . 6  |-  ( ph  ->  I  e.  B )
363, 18, 19, 20lmodvscl 15479 . . . . . 6  |-  ( ( U  e.  LMod  /\  I  e.  B  /\  E  e.  V )  ->  (
I  .x.  E )  e.  V )
379, 35, 17, 36syl3anc 1187 . . . . 5  |-  ( ph  ->  ( I  .x.  E
)  e.  V )
38 hdmapinvlem3.c . . . . . 6  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
3926, 38sseldd 3104 . . . . 5  |-  ( ph  ->  C  e.  V )
40 hdmapinvlem3.p . . . . . 6  |-  .+  =  ( +g  `  U )
413, 40lmodvacl 15476 . . . . 5  |-  ( ( U  e.  LMod  /\  (
I  .x.  E )  e.  V  /\  C  e.  V )  ->  (
( I  .x.  E
)  .+  C )  e.  V )
429, 37, 39, 41syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( I  .x.  E )  .+  C
)  e.  V )
431, 2, 3, 18, 31, 5, 32, 6, 8, 33, 34, 42lcdvsubval 30609 . . 3  |-  ( ph  ->  ( ( ( S `
 ( J  .x.  E ) ) (
-g `  ( (LCDual `  K ) `  W
) ) ( S `
 D ) ) `
 ( ( I 
.x.  E )  .+  C ) )  =  ( ( ( S `
 ( J  .x.  E ) ) `  ( ( I  .x.  E )  .+  C
) ) ( -g `  R ) ( ( S `  D ) `
 ( ( I 
.x.  E )  .+  C ) ) ) )
44 eqid 2253 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
451, 2, 3, 40, 18, 44, 7, 8, 37, 39, 22hdmaplna1 30901 . . . . 5  |-  ( ph  ->  ( ( S `  ( J  .x.  E ) ) `  ( ( I  .x.  E ) 
.+  C ) )  =  ( ( ( S `  ( J 
.x.  E ) ) `
 ( I  .x.  E ) ) ( +g  `  R ) ( ( S `  ( J  .x.  E ) ) `  C ) ) )
46 hdmapinvlem3.t . . . . . . . 8  |-  .X.  =  ( .r `  R )
47 hdmapinvlem3.g . . . . . . . 8  |-  G  =  ( (HGMap `  K
) `  W )
481, 2, 3, 19, 18, 20, 46, 7, 47, 8, 37, 17, 10hdmapglnm2 30905 . . . . . . 7  |-  ( ph  ->  ( ( S `  ( J  .x.  E ) ) `  ( I 
.x.  E ) )  =  ( ( ( S `  E ) `
 ( I  .x.  E ) )  .X.  ( G `  J ) ) )
491, 2, 3, 19, 18, 20, 46, 7, 8, 17, 17, 35hdmaplnm1 30903 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  E ) `  (
I  .x.  E )
)  =  ( I 
.X.  ( ( S `
 E ) `  E ) ) )
50 eqid 2253 . . . . . . . . . . 11  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
51 eqid 2253 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
521, 14, 50, 7, 8, 2, 18, 51hdmapevec2 30830 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  E ) `  E
)  =  ( 1r
`  R ) )
5352oveq2d 5726 . . . . . . . . 9  |-  ( ph  ->  ( I  .X.  (
( S `  E
) `  E )
)  =  ( I 
.X.  ( 1r `  R ) ) )
5418lmodrng 15470 . . . . . . . . . . 11  |-  ( U  e.  LMod  ->  R  e. 
Ring )
559, 54syl 17 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
5620, 46, 51rngridm 15200 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  B )  ->  (
I  .X.  ( 1r `  R ) )  =  I )
5755, 35, 56syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( I  .X.  ( 1r `  R ) )  =  I )
5849, 53, 573eqtrd 2289 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  (
I  .x.  E )
)  =  I )
5958oveq1d 5725 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  ( I  .x.  E ) )  .X.  ( G `  J ) )  =  ( I  .X.  ( G `  J )
) )
6048, 59eqtrd 2285 . . . . . 6  |-  ( ph  ->  ( ( S `  ( J  .x.  E ) ) `  ( I 
.x.  E ) )  =  ( I  .X.  ( G `  J ) ) )
611, 2, 3, 19, 18, 20, 46, 7, 47, 8, 39, 17, 10hdmapglnm2 30905 . . . . . . 7  |-  ( ph  ->  ( ( S `  ( J  .x.  E ) ) `  C )  =  ( ( ( S `  E ) `
 C )  .X.  ( G `  J ) ) )
62 hdmapinvlem3.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
631, 14, 24, 2, 3, 18, 20, 46, 62, 7, 8, 38hdmapinvlem1 30912 . . . . . . . 8  |-  ( ph  ->  ( ( S `  E ) `  C
)  =  .0.  )
6463oveq1d 5725 . . . . . . 7  |-  ( ph  ->  ( ( ( S `
 E ) `  C )  .X.  ( G `  J )
)  =  (  .0.  .X.  ( G `  J
) ) )
651, 2, 18, 20, 47, 8, 10hgmapcl 30883 . . . . . . . 8  |-  ( ph  ->  ( G `  J
)  e.  B )
6620, 46, 62rnglz 15212 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  J )  e.  B )  ->  (  .0.  .X.  ( G `  J ) )  =  .0.  )
6755, 65, 66syl2anc 645 . . . . . . 7  |-  ( ph  ->  (  .0.  .X.  ( G `  J )
)  =  .0.  )
6861, 64, 673eqtrd 2289 . . . . . 6  |-  ( ph  ->  ( ( S `  ( J  .x.  E ) ) `  C )  =  .0.  )
6960, 68oveq12d 5728 . . . . 5  |-  ( ph  ->  ( ( ( S `
 ( J  .x.  E ) ) `  ( I  .x.  E ) ) ( +g  `  R
) ( ( S `
 ( J  .x.  E ) ) `  C ) )  =  ( ( I  .X.  ( G `  J ) ) ( +g  `  R
)  .0.  ) )
70 rnggrp 15181 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
7155, 70syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
7218, 20, 46lmodmcl 15474 . . . . . . 7  |-  ( ( U  e.  LMod  /\  I  e.  B  /\  ( G `  J )  e.  B )  ->  (
I  .X.  ( G `  J ) )  e.  B )
739, 35, 65, 72syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( I  .X.  ( G `  J )
)  e.  B )
7420, 44, 62grprid 14348 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( I  .X.  ( G `
 J ) )  e.  B )  -> 
( ( I  .X.  ( G `  J ) ) ( +g  `  R
)  .0.  )  =  ( I  .X.  ( G `  J )
) )
7571, 73, 74syl2anc 645 . . . . 5  |-  ( ph  ->  ( ( I  .X.  ( G `  J ) ) ( +g  `  R
)  .0.  )  =  ( I  .X.  ( G `  J )
) )
7645, 69, 753eqtrd 2289 . . . 4  |-  ( ph  ->  ( ( S `  ( J  .x.  E ) ) `  ( ( I  .x.  E ) 
.+  C ) )  =  ( I  .X.  ( G `  J ) ) )
771, 2, 3, 40, 18, 44, 7, 8, 37, 39, 28hdmaplna1 30901 . . . . 5  |-  ( ph  ->  ( ( S `  D ) `  (
( I  .x.  E
)  .+  C )
)  =  ( ( ( S `  D
) `  ( I  .x.  E ) ) ( +g  `  R ) ( ( S `  D ) `  C
) ) )
781, 2, 3, 19, 18, 20, 46, 7, 8, 17, 28, 35hdmaplnm1 30903 . . . . . . 7  |-  ( ph  ->  ( ( S `  D ) `  (
I  .x.  E )
)  =  ( I 
.X.  ( ( S `
 D ) `  E ) ) )
791, 14, 24, 2, 3, 18, 20, 46, 62, 7, 8, 27hdmapinvlem2 30913 . . . . . . . 8  |-  ( ph  ->  ( ( S `  D ) `  E
)  =  .0.  )
8079oveq2d 5726 . . . . . . 7  |-  ( ph  ->  ( I  .X.  (
( S `  D
) `  E )
)  =  ( I 
.X.  .0.  ) )
8120, 46, 62rngrz 15213 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  I  e.  B )  ->  (
I  .X.  .0.  )  =  .0.  )
8255, 35, 81syl2anc 645 . . . . . . 7  |-  ( ph  ->  ( I  .X.  .0.  )  =  .0.  )
8378, 80, 823eqtrrd 2290 . . . . . 6  |-  ( ph  ->  .0.  =  ( ( S `  D ) `
 ( I  .x.  E ) ) )
84 hdmapinvlem3.ij . . . . . 6  |-  ( ph  ->  ( I  .X.  ( G `  J )
)  =  ( ( S `  D ) `
 C ) )
8583, 84oveq12d 5728 . . . . 5  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( I 
.X.  ( G `  J ) ) )  =  ( ( ( S `  D ) `
 ( I  .x.  E ) ) ( +g  `  R ) ( ( S `  D ) `  C
) ) )
8620, 44, 62grplid 14347 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( I  .X.  ( G `
 J ) )  e.  B )  -> 
(  .0.  ( +g  `  R ) ( I 
.X.  ( G `  J ) ) )  =  ( I  .X.  ( G `  J ) ) )
8771, 73, 86syl2anc 645 . . . . 5  |-  ( ph  ->  (  .0.  ( +g  `  R ) ( I 
.X.  ( G `  J ) ) )  =  ( I  .X.  ( G `  J ) ) )
8877, 85, 873eqtr2d 2291 . . . 4  |-  ( ph  ->  ( ( S `  D ) `  (
( I  .x.  E
)  .+  C )
)  =  ( I 
.X.  ( G `  J ) ) )
8976, 88oveq12d 5728 . . 3  |-  ( ph  ->  ( ( ( S `
 ( J  .x.  E ) ) `  ( ( I  .x.  E )  .+  C
) ) ( -g `  R ) ( ( S `  D ) `
 ( ( I 
.x.  E )  .+  C ) ) )  =  ( ( I 
.X.  ( G `  J ) ) (
-g `  R )
( I  .X.  ( G `  J )
) ) )
9043, 89eqtrd 2285 . 2  |-  ( ph  ->  ( ( ( S `
 ( J  .x.  E ) ) (
-g `  ( (LCDual `  K ) `  W
) ) ( S `
 D ) ) `
 ( ( I 
.x.  E )  .+  C ) )  =  ( ( I  .X.  ( G `  J ) ) ( -g `  R
) ( I  .X.  ( G `  J ) ) ) )
9120, 62, 31grpsubid 14385 . . 3  |-  ( ( R  e.  Grp  /\  ( I  .X.  ( G `
 J ) )  e.  B )  -> 
( ( I  .X.  ( G `  J ) ) ( -g `  R
) ( I  .X.  ( G `  J ) ) )  =  .0.  )
9271, 73, 91syl2anc 645 . 2  |-  ( ph  ->  ( ( I  .X.  ( G `  J ) ) ( -g `  R
) ( I  .X.  ( G `  J ) ) )  =  .0.  )
9330, 90, 923eqtrd 2289 1  |-  ( ph  ->  ( ( S `  ( ( J  .x.  E )  .-  D
) ) `  (
( I  .x.  E
)  .+  C )
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3075    C_ wss 3078   {csn 3544   <.cop 3547    _I cid 4197    |` cres 4582   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   .rcmulr 13083  Scalarcsca 13085   .scvsca 13086   0gc0g 13274   Grpcgrp 14197   -gcsg 14200   Ringcrg 15172   1rcur 15174   LModclmod 15462   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   DVecHcdvh 30069   ocHcoch 30338  LCDualclcd 30577  HVMapchvm 30747  HDMapchdma 30784  HGMapchg 30877
This theorem is referenced by:  hdmapinvlem4  30915
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-ot 3554  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 27967  df-lshyp 27968  df-lcv 28010  df-lfl 28049  df-lkr 28077  df-ldual 28115  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149  df-tgrp 29733  df-tendo 29745  df-edring 29747  df-dveca 29993  df-disoa 30020  df-dvech 30070  df-dib 30130  df-dic 30164  df-dih 30220  df-doch 30339  df-djh 30386  df-lcdual 30578  df-mapd 30616  df-hvmap 30748  df-hdmap1 30785  df-hdmap 30786  df-hgmap 30878
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