Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1l6lem2 Unicode version

Theorem hdmap1l6lem2 30800
Description: Lemma for hdmap1l6 30813. 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 2253 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 30101 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6e.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3215 . . . . . . 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 15569 . . . . . 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 3215 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
1614, 15syl 17 . . . . . 6  |-  ( ph  ->  Z  e.  V )
1710, 4, 11lspsncl 15569 . . . . . 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 2253 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
204, 19lsmcl 15671 . . . . 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 3215 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2422, 23syl 17 . . . . . . 7  |-  ( ph  ->  X  e.  V )
25 hdmap1l6.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
2610, 25lmodvacl 15476 . . . . . . . 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 15505 . . . . . . 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 15569 . . . . . 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 15569 . . . . . 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 15671 . . . . 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 30653 . . 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 2253 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
401, 2, 3, 4, 19, 38, 39, 5, 13, 18mapdlsm 30655 . . . . 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 30100 . . . . . . . . . . . . 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 15723 . . . . . . . . . . . 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 30796 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5541, 54eqeltrrd 2328 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
561, 3, 10, 28, 42, 11, 38, 43, 44, 45, 2, 46, 5, 22, 47, 7, 55, 53, 48hdmap1eq 30793 . . . . . . . 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 15721 . . . . . . . . . . . 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 30796 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6359, 62eqeltrrd 2328 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
641, 3, 10, 28, 42, 11, 38, 43, 44, 45, 2, 46, 5, 22, 47, 14, 63, 61, 48hdmap1eq 30793 . . . . . . . 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 5728 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { G } ) (
LSSum `  C ) ( L `  { E } ) ) )
6840, 67eqtrd 2285 . . . 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 30655 . . . . 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 30799 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
7372, 48oveq12d 5728 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7469, 73eqtrd 2285 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( L `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( L `  { F } ) ) )
7568, 74ineq12d 3279 . . 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 2285 . 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 30706 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7877fveq2d 5381 . 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 30582 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 10, 11, 38, 43, 45, 5, 47, 48, 24, 9, 55, 58, 16, 63, 66, 51mapdindp 30662 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
811, 2, 3, 10, 11, 38, 43, 45, 5, 55, 58, 9, 16, 63, 66, 50mapdncol 30661 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
821, 2, 3, 10, 11, 38, 43, 45, 5, 55, 58, 42, 71, 7mapdn0 30660 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 10, 11, 38, 43, 45, 5, 63, 66, 42, 71, 14mapdn0 30660 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8443, 44, 71, 39, 45, 79, 47, 80, 81, 82, 83, 70baerlem5b 30706 . 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 2295 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 2412    \ cdif 3075    i^i cin 3077   {csn 3544   {cpr 3545   <.cotp 3548   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   0gc0g 13274   -gcsg 14200   LSSumclsm 14780   LModclmod 15462   LSubSpclss 15524   LSpanclspn 15563   HLchlt 28341   LHypclh 28974   DVecHcdvh 30069  LCDualclcd 30577  mapdcmpd 30615  HDMap1chdma1 30783
This theorem is referenced by:  hdmap1l6a  30801
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-hdmap1 30785
  Copyright terms: Public domain W3C validator