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

Theorem hdmap1l6lem1 31266
Description: Lemma for hdmap1l6 31280. 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 2285 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 30568 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 hdmap1l6cl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3300 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
97, 8syl 17 . . . . . . 7  |-  ( ph  ->  X  e.  V )
10 hdmap1l6e.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3300 . . . . . . . 8  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1210, 11syl 17 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
13 hdmap1l6.v . . . . . . . 8  |-  V  =  ( Base `  U
)
14 hdmap1l6.s . . . . . . . 8  |-  .-  =  ( -g `  U )
1513, 14lmodvsubcl 15665 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
166, 9, 12, 15syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
17 hdmap1l6.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1813, 4, 17lspsncl 15729 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
196, 16, 18syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
20 hdmap1l6e.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
21 eldifi 3300 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
2220, 21syl 17 . . . . . 6  |-  ( ph  ->  Z  e.  V )
2313, 4, 17lspsncl 15729 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
246, 22, 23syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
25 eqid 2285 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
264, 25lsmcl 15831 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  e.  ( LSubSp `  U )
)
2813, 14lmodvsubcl 15665 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
296, 9, 22, 28syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
3013, 4, 17lspsncl 15729 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
316, 29, 30syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
3213, 4, 17lspsncl 15729 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
336, 12, 32syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
344, 25lsmcl 15831 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) )  e.  ( LSubSp `  U )
)
361, 2, 3, 4, 5, 27, 35mapdin 31120 . . 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 2285 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
391, 2, 3, 4, 25, 37, 38, 5, 19, 24mapdlsm 31122 . . . . 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 31122 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Z
) } ) (
LSSum `  U ) ( N `  { Y } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )
4139, 40ineq12d 3373 . . . 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 30567 . . . . . . . . . . . . 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 15883 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5453simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
551, 3, 13, 43, 17, 37, 44, 46, 2, 47, 5, 48, 49, 54, 7, 12hdmap1cl 31263 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5642, 55eqeltrrd 2360 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
571, 3, 13, 14, 43, 17, 37, 44, 45, 46, 2, 47, 5, 7, 48, 10, 56, 54, 49hdmap1eq 31260 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) ) )
5842, 57mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R G ) } ) ) )
5958simprd 451 . . . . . 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 15881 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6261simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
631, 3, 13, 43, 17, 37, 44, 46, 2, 47, 5, 48, 49, 62, 7, 22hdmap1cl 31263 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6460, 63eqeltrrd 2360 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
651, 3, 13, 14, 43, 17, 37, 44, 45, 46, 2, 47, 5, 7, 48, 20, 64, 62, 49hdmap1eq 31260 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) ) )
6660, 65mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( L `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `
 { ( F R E ) } ) ) )
6766simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( L `  { E } ) )
6859, 67oveq12d 5838 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( L `  { ( F R G ) } ) ( LSSum `  C )
( L `  { E } ) ) )
6966simprd 451 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( L `  { ( F R E ) } ) )
7058simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
7169, 70oveq12d 5838 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Z ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  =  ( ( L `  { ( F R E ) } ) ( LSSum `  C )
( L `  { G } ) ) )
7268, 71ineq12d 3373 . . . 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 2317 . . 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 2317 . 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 31172 . . 3  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )
7776fveq2d 5490 . 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 31049 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 13, 17, 37, 44, 46, 5, 48, 49, 9, 12, 56, 70, 22, 64, 67, 52mapdindp 31129 . . 3  |-  ( ph  ->  -.  F  e.  ( L `  { G ,  E } ) )
811, 2, 3, 13, 17, 37, 44, 46, 5, 56, 70, 12, 22, 64, 67, 51mapdncol 31128 . . 3  |-  ( ph  ->  ( L `  { G } )  =/=  ( L `  { E } ) )
821, 2, 3, 13, 17, 37, 44, 46, 5, 56, 70, 43, 78, 10mapdn0 31127 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 13, 17, 37, 44, 46, 5, 64, 67, 43, 78, 20mapdn0 31127 . . 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 31172 . 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 2327 1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G 
.+b  E ) ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448    \ cdif 3151    i^i cin 3153   {csn 3642   {cpr 3643   <.cotp 3646   ` cfv 5222  (class class class)co 5820   Basecbs 13143   +g cplusg 13203   0gc0g 13395   -gcsg 14360   LSSumclsm 14940   LModclmod 15622   LSubSpclss 15684   LSpanclspn 15723   HLchlt 28808   LHypclh 29441   DVecHcdvh 30536  LCDualclcd 31044  mapdcmpd 31082  HDMap1chdma1 31250
This theorem is referenced by:  hdmap1l6lem2  31267  hdmap1l6a  31268
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-undef 6292  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10778  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-sca 13219  df-vsca 13220  df-0g 13399  df-mre 13483  df-mrc 13484  df-acs 13486  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-mnd 14362  df-submnd 14411  df-grp 14484  df-minusg 14485  df-sbg 14486  df-subg 14613  df-cntz 14788  df-oppg 14814  df-lsm 14942  df-cmn 15086  df-abl 15087  df-mgp 15321  df-rng 15335  df-ur 15337  df-oppr 15400  df-dvdsr 15418  df-unit 15419  df-invr 15449  df-dvr 15460  df-drng 15509  df-lmod 15624  df-lss 15685  df-lsp 15724  df-lvec 15851  df-lsatoms 28434  df-lshyp 28435  df-lcv 28477  df-lfl 28516  df-lkr 28544  df-ldual 28582  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616  df-tgrp 30200  df-tendo 30212  df-edring 30214  df-dveca 30460  df-disoa 30487  df-dvech 30537  df-dib 30597  df-dic 30631  df-dih 30687  df-doch 30806  df-djh 30853  df-lcdual 31045  df-mapd 31083  df-hdmap1 31252
  Copyright terms: Public domain W3C validator