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Theorem hdmap1l6lem2 31925
Description: Lemma for hdmap1l6 31938. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.)
Hypotheses
Ref Expression
hdmap1l6.h  |-  H  =  ( LHyp `  K
)
hdmap1l6.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1l6.v  |-  V  =  ( Base `  U
)
hdmap1l6.p  |-  .+  =  ( +g  `  U )
hdmap1l6.s  |-  .-  =  ( -g `  U )
hdmap1l6c.o  |-  .0.  =  ( 0g `  U )
hdmap1l6.n  |-  N  =  ( LSpan `  U )
hdmap1l6.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1l6.d  |-  D  =  ( Base `  C
)
hdmap1l6.a  |-  .+b  =  ( +g  `  C )
hdmap1l6.r  |-  R  =  ( -g `  C
)
hdmap1l6.q  |-  Q  =  ( 0g `  C
)
hdmap1l6.l  |-  L  =  ( LSpan `  C )
hdmap1l6.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1l6.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1l6.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1l6.f  |-  ( ph  ->  F  e.  D )
hdmap1l6cl.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap1l6.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
hdmap1l6e.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap1l6e.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
hdmap1l6e.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
hdmap1l6.yz  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
hdmap1l6.fg  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
hdmap1l6.fe  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
Assertion
Ref Expression
hdmap1l6lem2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( L `  { ( G  .+b  E ) } ) )

Proof of Theorem hdmap1l6lem2
StepHypRef Expression
1 hdmap1l6.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmap1l6.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
3 hdmap1l6.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 eqid 2388 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31226 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6e.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
87eldifad 3276 . . . . . 6  |-  ( ph  ->  Y  e.  V )
9 hdmap1l6.v . . . . . . 7  |-  V  =  ( Base `  U
)
10 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
119, 4, 10lspsncl 15981 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
126, 8, 11syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
13 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1413eldifad 3276 . . . . . 6  |-  ( ph  ->  Z  e.  V )
159, 4, 10lspsncl 15981 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
166, 14, 15syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
17 eqid 2388 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
184, 17lsmcl 16083 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
196, 12, 16, 18syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
20 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3276 . . . . . . 7  |-  ( ph  ->  X  e.  V )
22 hdmap1l6.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
239, 22lmodvacl 15892 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
246, 8, 14, 23syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
25 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
269, 25lmodvsubcl 15917 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
276, 21, 24, 26syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
289, 4, 10lspsncl 15981 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
296, 27, 28syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
309, 4, 10lspsncl 15981 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
316, 21, 30syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
324, 17lsmcl 16083 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U )  /\  ( N `  { X } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) )  e.  ( LSubSp `  U
) )
336, 29, 31, 32syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
341, 2, 3, 4, 5, 19, 33mapdin 31778 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( M `  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
35 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
36 eqid 2388 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 31780 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
38 hdmap1l6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
39 hdmap1l6c.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
40 hdmap1l6.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
41 hdmap1l6.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
42 hdmap1l6.l . . . . . . . . 9  |-  L  =  ( LSpan `  C )
43 hdmap1l6.i . . . . . . . . 9  |-  I  =  ( (HDMap1 `  K
) `  W )
44 hdmap1l6.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
45 hdmap1l6.mn . . . . . . . . . . 11  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
461, 3, 5dvhlvec 31225 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
47 hdmap1l6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
48 hdmap1l6e.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
499, 39, 10, 46, 8, 13, 21, 47, 48lspindp2 16135 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5049simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
511, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 50, 20, 8hdmap1cl 31921 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5238, 51eqeltrrd 2463 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
531, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 7, 52, 50, 45hdmap1eq 31918 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5438, 53mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5554simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
56 hdmap1l6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
579, 39, 10, 46, 7, 14, 21, 47, 48lspindp1 16133 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5857simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
591, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 58, 20, 14hdmap1cl 31921 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6056, 59eqeltrrd 2463 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
611, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 13, 60, 58, 45hdmap1eq 31918 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6256, 61mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6362simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6455, 63oveq12d 6039 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
6537, 64eqtrd 2420 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
661, 2, 3, 4, 17, 35, 36, 5, 29, 31mapdlsm 31780 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) ) )
67 hdmap1l6.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
68 hdmap1l6.q . . . . . . 7  |-  Q  =  ( 0g `  C
)
691, 3, 9, 22, 25, 39, 10, 35, 40, 67, 41, 68, 42, 2, 43, 5, 44, 20, 45, 7, 13, 48, 47, 38, 56hdmap1l6lem1 31924 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
7069, 45oveq12d 6039 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7166, 70eqtrd 2420 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7265, 71ineq12d 3487 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( L `  { G } ) ( LSSum `  C ) ( L `
 { E }
) )  i^i  (
( L `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( L `  { F } ) ) ) )
7334, 72eqtrd 2420 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( L `  { G } ) ( LSSum `  C ) ( L `
 { E }
) )  i^i  (
( L `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( L `  { F } ) ) ) )
749, 25, 39, 17, 10, 46, 21, 48, 47, 7, 13, 22baerlem5b 31831 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7574fveq2d 5673 . 2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( M `  ( ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
761, 35, 5lcdlvec 31707 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 9, 10, 35, 40, 42, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 31787 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
781, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 31786 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
791, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 39, 68, 7mapdn0 31785 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13mapdn0 31785 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8140, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67baerlem5b 31831 . 2  |-  ( ph  ->  ( L `  {
( G  .+b  E
) } )  =  ( ( ( L `
 { G }
) ( LSSum `  C
) ( L `  { E } ) )  i^i  ( ( L `
 { ( F R ( G  .+b  E ) ) } ) ( LSSum `  C )
( L `  { F } ) ) ) )
8273, 75, 813eqtr4d 2430 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( L `  { ( G  .+b  E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551    \ cdif 3261    i^i cin 3263   {csn 3758   {cpr 3759   <.cotp 3762   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   0gc0g 13651   -gcsg 14616   LSSumclsm 15196   LModclmod 15878   LSubSpclss 15936   LSpanclspn 15975   HLchlt 29466   LHypclh 30099   DVecHcdvh 31194  LCDualclcd 31702  mapdcmpd 31740  HDMap1chdma1 31908
This theorem is referenced by:  hdmap1l6a  31926
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-ot 3768  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-tpos 6416  df-undef 6480  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-0g 13655  df-mre 13739  df-mrc 13740  df-acs 13742  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-mnd 14618  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-cntz 15044  df-oppg 15070  df-lsm 15198  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-dvr 15716  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103  df-lsatoms 29092  df-lshyp 29093  df-lcv 29135  df-lfl 29174  df-lkr 29202  df-ldual 29240  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274  df-tgrp 30858  df-tendo 30870  df-edring 30872  df-dveca 31118  df-disoa 31145  df-dvech 31195  df-dib 31255  df-dic 31289  df-dih 31345  df-doch 31464  df-djh 31511  df-lcdual 31703  df-mapd 31741  df-hdmap1 31910
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