Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1l6lem1 Unicode version

Theorem hdmap1l6lem1 32291
Description: Lemma for hdmap1l6 32305. 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 2404 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31593 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
87eldifad 3292 . . . . . . 7  |-  ( ph  ->  X  e.  V )
9 hdmap1l6e.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
109eldifad 3292 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
11 hdmap1l6.v . . . . . . . 8  |-  V  =  ( Base `  U
)
12 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
1311, 12lmodvsubcl 15944 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
146, 8, 10, 13syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
15 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1611, 4, 15lspsncl 16008 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
176, 14, 16syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
18 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
1918eldifad 3292 . . . . . 6  |-  ( ph  ->  Z  e.  V )
2011, 4, 15lspsncl 16008 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
216, 19, 20syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
22 eqid 2404 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
234, 22lsmcl 16110 . . . . 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
) )
246, 17, 21, 23syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  e.  ( LSubSp `  U )
)
2511, 12lmodvsubcl 15944 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
266, 8, 19, 25syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
2711, 4, 15lspsncl 16008 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
286, 26, 27syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
2911, 4, 15lspsncl 16008 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
306, 10, 29syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
314, 22lsmcl 16110 . . . . 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
) )
326, 28, 30, 31syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) )  e.  ( LSubSp `  U )
)
331, 2, 3, 4, 5, 24, 32mapdin 32145 . . 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 } ) ) ) ) )
34 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
35 eqid 2404 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
361, 2, 3, 4, 22, 34, 35, 5, 17, 21mapdlsm 32147 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) ) )
371, 2, 3, 4, 22, 34, 35, 5, 28, 30mapdlsm 32147 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )
3836, 37ineq12d 3503 . . . 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 } ) ) ) ) )
39 hdmap1l6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
40 hdmap1l6c.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
41 hdmap1l6.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
42 hdmap1l6.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
43 hdmap1l6.l . . . . . . . . 9  |-  L  =  ( LSpan `  C )
44 hdmap1l6.i . . . . . . . . 9  |-  I  =  ( (HDMap1 `  K
) `  W )
45 hdmap1l6.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
46 hdmap1l6.mn . . . . . . . . . . 11  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
471, 3, 5dvhlvec 31592 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
48 hdmap1l6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
49 hdmap1l6e.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5011, 40, 15, 47, 10, 18, 8, 48, 49lspindp2 16162 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5150simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
521, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 51, 7, 10hdmap1cl 32288 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5339, 52eqeltrrd 2479 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
541, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 9, 53, 51, 46hdmap1eq 32285 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5539, 54mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5655simprd 450 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `  { ( F R G ) } ) )
57 hdmap1l6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
5811, 40, 15, 47, 9, 19, 8, 48, 49lspindp1 16160 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
5958simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
601, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 59, 7, 19hdmap1cl 32288 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6157, 60eqeltrrd 2479 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
621, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 18, 61, 59, 46hdmap1eq 32285 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6357, 62mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6463simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6556, 64oveq12d 6058 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { ( F R G ) } ) ( LSSum `  C )
( L `  { E } ) ) )
6663simprd 450 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `  { ( F R E ) } ) )
6755simpld 446 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
6866, 67oveq12d 6058 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Z ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  =  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) )
6965, 68ineq12d 3503 . . . 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 } ) ) ) )
7038, 69eqtrd 2436 . . 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 } ) ) ) )
7133, 70eqtrd 2436 . 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 } ) ) ) )
72 hdmap1l6.p . . . 4  |-  .+  =  ( +g  `  U )
7311, 12, 40, 22, 15, 47, 8, 49, 48, 9, 18, 72baerlem5a 32197 . . 3  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )
7473fveq2d 5691 . 2  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( M `  ( ( ( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) ) )
75 hdmap1l6.q . . 3  |-  Q  =  ( 0g `  C
)
761, 34, 5lcdlvec 32074 . . 3  |-  ( ph  ->  C  e.  LVec )
771, 2, 3, 11, 15, 34, 41, 43, 5, 45, 46, 8, 10, 53, 67, 19, 61, 64, 49mapdindp 32154 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
781, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 10, 19, 61, 64, 48mapdncol 32153 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
791, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 40, 75, 9mapdn0 32152 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
801, 2, 3, 11, 15, 34, 41, 43, 5, 61, 64, 40, 75, 18mapdn0 32152 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
81 hdmap1l6.a . . 3  |-  .+b  =  ( +g  `  C )
8241, 42, 75, 35, 43, 76, 45, 77, 78, 79, 80, 81baerlem5a 32197 . 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 } ) ) ) )
8371, 74, 823eqtr4d 2446 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 359    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    i^i cin 3279   {csn 3774   {cpr 3775   <.cotp 3778   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   -gcsg 14643   LSSumclsm 15223   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561  LCDualclcd 32069  mapdcmpd 32107  HDMap1chdma1 32275
This theorem is referenced by:  hdmap1l6lem2  32292  hdmap1l6a  32293
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lcv 29502  df-lfl 29541  df-lkr 29569  df-ldual 29607  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-lcdual 32070  df-mapd 32108  df-hdmap1 32277
  Copyright terms: Public domain W3C validator