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Theorem hdmap1l6lem2 31249
Description: Lemma for hdmap1l6 31262. 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 2258 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 30550 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6e.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3273 . . . . . . 7  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
97, 8syl 17 . . . . . 6  |-  ( ph  ->  Y  e.  V )
10 hdmap1l6.v . . . . . . 7  |-  V  =  ( Base `  U
)
11 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1210, 4, 11lspsncl 15697 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
136, 9, 12syl2anc 645 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
14 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
15 eldifi 3273 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
1614, 15syl 17 . . . . . 6  |-  ( ph  ->  Z  e.  V )
1710, 4, 11lspsncl 15697 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
186, 16, 17syl2anc 645 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
19 eqid 2258 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
204, 19lsmcl 15799 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
216, 13, 18, 20syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
22 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
23 eldifi 3273 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2422, 23syl 17 . . . . . . 7  |-  ( ph  ->  X  e.  V )
25 hdmap1l6.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
2610, 25lmodvacl 15604 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
276, 9, 16, 26syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
28 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
2910, 28lmodvsubcl 15633 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
306, 24, 27, 29syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
3110, 4, 11lspsncl 15697 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
326, 30, 31syl2anc 645 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
3310, 4, 11lspsncl 15697 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
346, 24, 33syl2anc 645 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
354, 19lsmcl 15799 . . . . 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
) )
366, 32, 34, 35syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
371, 2, 3, 4, 5, 21, 36mapdin 31102 . . 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 }
) ) ) ) )
38 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
39 eqid 2258 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
401, 2, 3, 4, 19, 38, 39, 5, 13, 18mapdlsm 31104 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
41 hdmap1l6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
42 hdmap1l6c.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
43 hdmap1l6.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
44 hdmap1l6.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
45 hdmap1l6.l . . . . . . . . 9  |-  L  =  ( LSpan `  C )
46 hdmap1l6.i . . . . . . . . 9  |-  I  =  ( (HDMap1 `  K
) `  W )
47 hdmap1l6.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
48 hdmap1l6.mn . . . . . . . . . . 11  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
491, 3, 5dvhlvec 30549 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
50 hdmap1l6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
51 hdmap1l6e.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5210, 42, 11, 49, 9, 14, 24, 50, 51lspindp2 15851 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5352simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
541, 3, 10, 42, 11, 38, 43, 45, 2, 46, 5, 47, 48, 53, 22, 9hdmap1cl 31245 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5541, 54eqeltrrd 2333 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
561, 3, 10, 28, 42, 11, 38, 43, 44, 45, 2, 46, 5, 22, 47, 7, 55, 53, 48hdmap1eq 31242 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5741, 56mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5857simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
59 hdmap1l6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
6010, 42, 11, 49, 7, 16, 24, 50, 51lspindp1 15849 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6160simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
621, 3, 10, 42, 11, 38, 43, 45, 2, 46, 5, 47, 48, 61, 22, 16hdmap1cl 31245 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6359, 62eqeltrrd 2333 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
641, 3, 10, 28, 42, 11, 38, 43, 44, 45, 2, 46, 5, 22, 47, 14, 63, 61, 48hdmap1eq 31242 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6559, 64mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6665simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6758, 66oveq12d 5810 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
6840, 67eqtrd 2290 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
691, 2, 3, 4, 19, 38, 39, 5, 32, 34mapdlsm 31104 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) ) )
70 hdmap1l6.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
71 hdmap1l6.q . . . . . . 7  |-  Q  =  ( 0g `  C
)
721, 3, 10, 25, 28, 42, 11, 38, 43, 70, 44, 71, 45, 2, 46, 5, 47, 22, 48, 7, 14, 51, 50, 41, 59hdmap1l6lem1 31248 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
7372, 48oveq12d 5810 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7469, 73eqtrd 2290 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7568, 74ineq12d 3346 . . 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 } ) ) ) )
7637, 75eqtrd 2290 . 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 } ) ) ) )
7710, 28, 42, 19, 11, 49, 24, 51, 50, 7, 14, 25baerlem5b 31155 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7877fveq2d 5462 . 2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( M `  ( ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) ) ) )
791, 38, 5lcdlvec 31031 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 10, 11, 38, 43, 45, 5, 47, 48, 24, 9, 55, 58, 16, 63, 66, 51mapdindp 31111 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
811, 2, 3, 10, 11, 38, 43, 45, 5, 55, 58, 9, 16, 63, 66, 50mapdncol 31110 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
821, 2, 3, 10, 11, 38, 43, 45, 5, 55, 58, 42, 71, 7mapdn0 31109 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 10, 11, 38, 43, 45, 5, 63, 66, 42, 71, 14mapdn0 31109 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8443, 44, 71, 39, 45, 79, 47, 80, 81, 82, 83, 70baerlem5b 31155 . 2  |-  ( ph  ->  ( L `  {
( G  .+b  E
) } )  =  ( ( ( L `
 { G }
) ( LSSum `  C
) ( L `  { E } ) )  i^i  ( ( L `
 { ( F R ( G  .+b  E ) ) } ) ( LSSum `  C )
( L `  { F } ) ) ) )
8576, 78, 843eqtr4d 2300 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( L `  { ( G  .+b  E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421    \ cdif 3124    i^i cin 3126   {csn 3614   {cpr 3615   <.cotp 3618   ` cfv 4673  (class class class)co 5792   Basecbs 13111   +g cplusg 13171   0gc0g 13363   -gcsg 14328   LSSumclsm 14908   LModclmod 15590   LSubSpclss 15652   LSpanclspn 15691   HLchlt 28790   LHypclh 29423   DVecHcdvh 30518  LCDualclcd 31026  mapdcmpd 31064  HDMap1chdma1 31232
This theorem is referenced by:  hdmap1l6a  31250
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9934  df-z 9993  df-uz 10199  df-fz 10750  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-sca 13187  df-vsca 13188  df-0g 13367  df-mre 13451  df-mrc 13452  df-acs 13454  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-mnd 14330  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-subg 14581  df-cntz 14756  df-oppg 14782  df-lsm 14910  df-cmn 15054  df-abl 15055  df-mgp 15289  df-ring 15303  df-ur 15305  df-oppr 15368  df-dvdsr 15386  df-unit 15387  df-invr 15417  df-dvr 15428  df-drng 15477  df-lmod 15592  df-lss 15653  df-lsp 15692  df-lvec 15819  df-lsatoms 28416  df-lshyp 28417  df-lcv 28459  df-lfl 28498  df-lkr 28526  df-ldual 28564  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598  df-tgrp 30182  df-tendo 30194  df-edring 30196  df-dveca 30442  df-disoa 30469  df-dvech 30519  df-dib 30579  df-dic 30613  df-dih 30669  df-doch 30788  df-djh 30835  df-lcdual 31027  df-mapd 31065  df-hdmap1 31234
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