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Theorem hdmap1l6lem1 31998
Description: Lemma for hdmap1l6 32012. Part (6) in [Baer] p. 47, lines 16-18. (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
hdmap1l6lem1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )

Proof of Theorem hdmap1l6lem1
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 2283 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31300 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3298 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  X  e.  V )
10 hdmap1l6e.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3298 . . . . . . . 8  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1210, 11syl 15 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
13 hdmap1l6.v . . . . . . . 8  |-  V  =  ( Base `  U
)
14 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
1513, 14lmodvsubcl 15670 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
166, 9, 12, 15syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
17 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1813, 4, 17lspsncl 15734 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
196, 16, 18syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
20 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
21 eldifi 3298 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
2220, 21syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  V )
2313, 4, 17lspsncl 15734 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
246, 22, 23syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
25 eqid 2283 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
264, 25lsmcl 15836 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U )  /\  ( N `  { Z } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { Z } ) )  e.  ( LSubSp `  U
) )
276, 19, 24, 26syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  e.  ( LSubSp `  U )
)
2813, 14lmodvsubcl 15670 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
296, 9, 22, 28syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
3013, 4, 17lspsncl 15734 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
316, 29, 30syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
3213, 4, 17lspsncl 15734 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
336, 12, 32syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
344, 25lsmcl 15836 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U )  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { ( X 
.-  Z ) } ) ( LSSum `  U
) ( N `  { Y } ) )  e.  ( LSubSp `  U
) )
356, 31, 33, 34syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) )  e.  ( LSubSp `  U )
)
361, 2, 3, 4, 5, 27, 35mapdin 31852 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) ) ) )
37 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
38 eqid 2283 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
391, 2, 3, 4, 25, 37, 38, 5, 19, 24mapdlsm 31854 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) ) )
401, 2, 3, 4, 25, 37, 38, 5, 31, 33mapdlsm 31854 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )
4139, 40ineq12d 3371 . . . 4  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( M `  ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C ) ( M `
 ( N `  { Z } ) ) )  i^i  ( ( M `  ( N `
 { ( X 
.-  Z ) } ) ) ( LSSum `  C ) ( M `
 ( N `  { Y } ) ) ) ) )
42 hdmap1l6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
43 hdmap1l6c.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
44 hdmap1l6.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
45 hdmap1l6.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
46 hdmap1l6.l . . . . . . . . 9  |-  L  =  ( LSpan `  C )
47 hdmap1l6.i . . . . . . . . 9  |-  I  =  ( (HDMap1 `  K
) `  W )
48 hdmap1l6.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
49 hdmap1l6.mn . . . . . . . . . . 11  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
501, 3, 5dvhlvec 31299 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
51 hdmap1l6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
52 hdmap1l6e.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5313, 43, 17, 50, 12, 20, 9, 51, 52lspindp2 15888 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5453simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
551, 3, 13, 43, 17, 37, 44, 46, 2, 47, 5, 48, 49, 54, 7, 12hdmap1cl 31995 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5642, 55eqeltrrd 2358 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
571, 3, 13, 14, 43, 17, 37, 44, 45, 46, 2, 47, 5, 7, 48, 10, 56, 54, 49hdmap1eq 31992 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5842, 57mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5958simprd 449 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R G ) } ) )
60 hdmap1l6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
6113, 43, 17, 50, 10, 22, 9, 51, 52lspindp1 15886 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6261simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
631, 3, 13, 43, 17, 37, 44, 46, 2, 47, 5, 48, 49, 62, 7, 22hdmap1cl 31995 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6460, 63eqeltrrd 2358 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
651, 3, 13, 14, 43, 17, 37, 44, 45, 46, 2, 47, 5, 7, 48, 20, 64, 62, 49hdmap1eq 31992 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6660, 65mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6766simpld 445 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6859, 67oveq12d 5876 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { ( F R G ) } ) ( LSSum `  C )
( L `  { E } ) ) )
6966simprd 449 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `  { ( F R E ) } ) )
7058simpld 445 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
7169, 70oveq12d 5876 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Z ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  =  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) )
7268, 71ineq12d 3371 . . . 4  |-  ( ph  ->  ( ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) )  i^i  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )  =  ( ( ( L `  {
( F R G ) } ) (
LSSum `  C ) ( L `  { E } ) )  i^i  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) ) )
7341, 72eqtrd 2315 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( L `  {
( F R G ) } ) (
LSSum `  C ) ( L `  { E } ) )  i^i  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) ) )
7436, 73eqtrd 2315 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( L `  {
( F R G ) } ) (
LSSum `  C ) ( L `  { E } ) )  i^i  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) ) )
75 hdmap1l6.p . . . 4  |-  .+  =  ( +g  `  U )
7613, 14, 43, 25, 17, 50, 9, 52, 51, 10, 20, 75baerlem5a 31904 . . 3  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )
7776fveq2d 5529 . 2  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( M `  ( ( ( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) ) )
78 hdmap1l6.q . . 3  |-  Q  =  ( 0g `  C
)
791, 37, 5lcdlvec 31781 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 13, 17, 37, 44, 46, 5, 48, 49, 9, 12, 56, 70, 22, 64, 67, 52mapdindp 31861 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
811, 2, 3, 13, 17, 37, 44, 46, 5, 56, 70, 12, 22, 64, 67, 51mapdncol 31860 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
821, 2, 3, 13, 17, 37, 44, 46, 5, 56, 70, 43, 78, 10mapdn0 31859 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 13, 17, 37, 44, 46, 5, 64, 67, 43, 78, 20mapdn0 31859 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
84 hdmap1l6.a . . 3  |-  .+b  =  ( +g  `  C )
8544, 45, 78, 38, 46, 79, 48, 80, 81, 82, 83, 84baerlem5a 31904 . 2  |-  ( ph  ->  ( L `  {
( F R ( G  .+b  E )
) } )  =  ( ( ( L `
 { ( F R G ) } ) ( LSSum `  C
) ( L `  { E } ) )  i^i  ( ( L `
 { ( F R E ) } ) ( LSSum `  C
) ( L `  { G } ) ) ) )
8674, 77, 853eqtr4d 2325 1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151   {csn 3640   {cpr 3641   <.cotp 3644   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   -gcsg 14365   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814  HDMap1chdma1 31982
This theorem is referenced by:  hdmap1l6lem2  31999  hdmap1l6a  32000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-lcdual 31777  df-mapd 31815  df-hdmap1 31984
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