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Theorem mapdh6lem2N 31191
Description: Lemma for mapdh6N 31204. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
mapdh.q  |-  Q  =  ( 0g `  C
)
mapdh.i  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
mapdh.h  |-  H  =  ( LHyp `  K
)
mapdh.m  |-  M  =  ( (mapd `  K
) `  W )
mapdh.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdh.v  |-  V  =  ( Base `  U
)
mapdh.s  |-  .-  =  ( -g `  U )
mapdhc.o  |-  .0.  =  ( 0g `  U )
mapdh.n  |-  N  =  ( LSpan `  U )
mapdh.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdh.d  |-  D  =  ( Base `  C
)
mapdh.r  |-  R  =  ( -g `  C
)
mapdh.j  |-  J  =  ( LSpan `  C )
mapdh.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdhc.f  |-  ( ph  ->  F  e.  D )
mapdh.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
mapdhcl.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
mapdh.p  |-  .+  =  ( +g  `  U )
mapdh.a  |-  .+b  =  ( +g  `  C )
mapdhe6.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdhe6.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
mapdhe6.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
mapdh6.yz  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
mapdh6.fg  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
mapdh6.fe  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
Assertion
Ref Expression
mapdh6lem2N  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( J `  { ( G  .+b  E ) } ) )
Distinct variable groups:    x, D, h    h, F, x    x, J    x, M    x, N    x,  .0.    x, Q    x, R    x, 
.-    h, X, x    h, Y, x    ph, h    .0. , h    C, h    D, h   
h, J    h, M    h, N    R, h    U, h    .- , h    h, G, x   
h, E    h, Z, x   
.+b , h    h, I    .+ , h, x
Allowed substitution hints:    ph( x)    C( x)   
.+b ( x)    Q( h)    U( x)    E( x)    H( x, h)    I( x)    K( x, h)    V( x, h)    W( x, h)

Proof of Theorem mapdh6lem2N
StepHypRef Expression
1 mapdh.h . . . 4  |-  H  =  ( LHyp `  K
)
2 mapdh.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
3 mapdh.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 eqid 2284 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 30567 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe6.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3299 . . . . . . 7  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
97, 8syl 17 . . . . . 6  |-  ( ph  ->  Y  e.  V )
10 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
11 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1210, 4, 11lspsncl 15728 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
136, 9, 12syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
14 mapdhe6.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
15 eldifi 3299 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
1614, 15syl 17 . . . . . 6  |-  ( ph  ->  Z  e.  V )
1710, 4, 11lspsncl 15728 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
186, 16, 17syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
19 eqid 2284 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
204, 19lsmcl 15830 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
22 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
23 eldifi 3299 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2422, 23syl 17 . . . . . . 7  |-  ( ph  ->  X  e.  V )
25 mapdh.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
2610, 25lmodvacl 15635 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
276, 9, 16, 26syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
28 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
2910, 28lmodvsubcl 15664 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
306, 24, 27, 29syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
3110, 4, 11lspsncl 15728 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
326, 30, 31syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
3310, 4, 11lspsncl 15728 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
346, 24, 33syl2anc 644 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
354, 19lsmcl 15830 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
371, 2, 3, 4, 5, 21, 36mapdin 31119 . . 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 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
39 eqid 2284 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
401, 2, 3, 4, 19, 38, 39, 5, 13, 18mapdlsm 31121 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
41 mapdh6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
42 mapdh.q . . . . . . . . 9  |-  Q  =  ( 0g `  C
)
43 mapdh.i . . . . . . . . 9  |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
44 mapdhc.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
45 mapdh.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
46 mapdh.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
47 mapdh.j . . . . . . . . 9  |-  J  =  ( LSpan `  C )
48 mapdhc.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
49 mapdh.mn . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
501, 3, 5dvhlvec 30566 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
51 mapdh6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
52 mapdhe6.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5310, 44, 11, 50, 9, 14, 24, 51, 52lspindp2 15882 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5453simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
5542, 43, 1, 2, 3, 10, 28, 44, 11, 38, 45, 46, 47, 5, 48, 49, 22, 9, 54mapdhcl 31184 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5641, 55eqeltrrd 2359 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
5742, 43, 1, 2, 3, 10, 28, 44, 11, 38, 45, 46, 47, 5, 48, 49, 22, 7, 56, 54mapdheq 31185 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5841, 57mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5958simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { G } ) )
60 mapdh6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
6110, 44, 11, 50, 7, 16, 24, 51, 52lspindp1 15880 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6261simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
6342, 43, 1, 2, 3, 10, 28, 44, 11, 38, 45, 46, 47, 5, 48, 49, 22, 16, 62mapdhcl 31184 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6460, 63eqeltrrd 2359 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
6542, 43, 1, 2, 3, 10, 28, 44, 11, 38, 45, 46, 47, 5, 48, 49, 22, 14, 64, 62mapdheq 31185 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6660, 65mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6766simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6859, 67oveq12d 5837 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6940, 68eqtrd 2316 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
701, 2, 3, 4, 19, 38, 39, 5, 32, 34mapdlsm 31121 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) ) )
71 mapdh.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
7242, 43, 1, 2, 3, 10, 28, 44, 11, 38, 45, 46, 47, 5, 48, 49, 22, 25, 71, 7, 14, 52, 51, 41, 60mapdh6lem1N 31190 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  E ) ) } ) )
7372, 49oveq12d 5837 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7470, 73eqtrd 2316 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7569, 74ineq12d 3372 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( J `  { G } ) ( LSSum `  C ) ( J `
 { E }
) )  i^i  (
( J `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( J `  { F } ) ) ) )
7637, 75eqtrd 2316 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) ) ) )  =  ( ( ( J `  { G } ) ( LSSum `  C ) ( J `
 { E }
) )  i^i  (
( J `  {
( F R ( G  .+b  E )
) } ) (
LSSum `  C ) ( J `  { F } ) ) ) )
7710, 28, 44, 19, 11, 50, 24, 52, 51, 7, 14, 25baerlem5b 31172 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7877fveq2d 5489 . 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 31048 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 10, 11, 38, 45, 47, 5, 48, 49, 24, 9, 56, 59, 16, 64, 67, 52mapdindp 31128 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
811, 2, 3, 10, 11, 38, 45, 47, 5, 56, 59, 9, 16, 64, 67, 51mapdncol 31127 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
821, 2, 3, 10, 11, 38, 45, 47, 5, 56, 59, 44, 42, 7mapdn0 31126 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 10, 11, 38, 45, 47, 5, 64, 67, 44, 42, 14mapdn0 31126 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8445, 46, 42, 39, 47, 79, 48, 80, 81, 82, 83, 71baerlem5b 31172 . 2  |-  ( ph  ->  ( J `  {
( G  .+b  E
) } )  =  ( ( ( J `
 { G }
) ( LSSum `  C
) ( J `  { E } ) )  i^i  ( ( J `
 { ( F R ( G  .+b  E ) ) } ) ( LSSum `  C )
( J `  { F } ) ) ) )
8576, 78, 843eqtr4d 2326 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( J `  { ( G  .+b  E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447   _Vcvv 2789    \ cdif 3150    i^i cin 3152   ifcif 3566   {csn 3641   {cpr 3642   <.cotp 3645    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082   iota_crio 6290   Basecbs 13142   +g cplusg 13202   0gc0g 13394   -gcsg 14359   LSSumclsm 14939   LModclmod 15621   LSubSpclss 15683   LSpanclspn 15722   HLchlt 28807   LHypclh 29440   DVecHcdvh 30535  LCDualclcd 31043  mapdcmpd 31081
This theorem is referenced by:  mapdh6aN  31192
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 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
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 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-ot 3651  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-undef 6291  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-n0 9961  df-z 10020  df-uz 10226  df-fz 10777  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-0g 13398  df-mre 13482  df-mrc 13483  df-acs 13485  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-mnd 14361  df-submnd 14410  df-grp 14483  df-minusg 14484  df-sbg 14485  df-subg 14612  df-cntz 14787  df-oppg 14813  df-lsm 14941  df-cmn 15085  df-abl 15086  df-mgp 15320  df-rng 15334  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-dvr 15459  df-drng 15508  df-lmod 15623  df-lss 15684  df-lsp 15723  df-lvec 15850  df-lsatoms 28433  df-lshyp 28434  df-lcv 28476  df-lfl 28515  df-lkr 28543  df-ldual 28581  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tgrp 30199  df-tendo 30211  df-edring 30213  df-dveca 30459  df-disoa 30486  df-dvech 30536  df-dib 30596  df-dic 30630  df-dih 30686  df-doch 30805  df-djh 30852  df-lcdual 31044  df-mapd 31082
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