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Theorem mapdh6lem2N 32546
Description: Lemma for mapdh6N 32559. 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 2296 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31922 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe6.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3311 . . . . . . 7  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
97, 8syl 15 . . . . . 6  |-  ( ph  ->  Y  e.  V )
10 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
11 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1210, 4, 11lspsncl 15750 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
136, 9, 12syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
14 mapdhe6.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
15 eldifi 3311 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
1614, 15syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  V )
1710, 4, 11lspsncl 15750 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
186, 16, 17syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
19 eqid 2296 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
204, 19lsmcl 15852 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  e.  ( LSubSp `  U )
)
22 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
23 eldifi 3311 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2422, 23syl 15 . . . . . . 7  |-  ( ph  ->  X  e.  V )
25 mapdh.p . . . . . . . . 9  |-  .+  =  ( +g  `  U )
2610, 25lmodvacl 15657 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
276, 9, 16, 26syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
28 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
2910, 28lmodvsubcl 15686 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  ( Y  .+  Z )  e.  V )  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
306, 24, 27, 29syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  Z ) )  e.  V )
3110, 4, 11lspsncl 15750 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  ( Y  .+  Z ) )  e.  V )  ->  ( N `  { ( X  .-  ( Y  .+  Z ) ) } )  e.  ( LSubSp `  U ) )
326, 30, 31syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  e.  (
LSubSp `  U ) )
3310, 4, 11lspsncl 15750 . . . . . 6  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
346, 24, 33syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
354, 19lsmcl 15852 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) (
LSSum `  U ) ( N `  { X } ) )  e.  ( LSubSp `  U )
)
371, 2, 3, 4, 5, 21, 36mapdin 32474 . . 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 2296 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
401, 2, 3, 4, 19, 38, 39, 5, 13, 18mapdlsm 32476 . . . . 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 31921 . . . . . . . . . . . . 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 15904 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5453simpld 445 . . . . . . . . . . 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 32539 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5641, 55eqeltrrd 2371 . . . . . . . . 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 32540 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5841, 57mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5958simpld 445 . . . . . 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 15902 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6261simpld 445 . . . . . . . . . . 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 32539 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6460, 63eqeltrrd 2371 . . . . . . . . 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 32540 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6660, 65mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6766simpld 445 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6859, 67oveq12d 5892 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6940, 68eqtrd 2328 . . . 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 32476 . . . . 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 32545 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  E ) ) } ) )
7372, 49oveq12d 5892 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  ( Y 
.+  Z ) ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7470, 73eqtrd 2328 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  ( Y  .+  Z ) ) } ) ( LSSum `  U ) ( N `
 { X }
) ) )  =  ( ( J `  { ( F R ( G  .+b  E
) ) } ) ( LSSum `  C )
( J `  { F } ) ) )
7569, 74ineq12d 3384 . . 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 2328 . 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 32527 . . 3  |-  ( ph  ->  ( N `  {
( Y  .+  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .+  Z ) ) } ) ( LSSum `  U
) ( N `  { X } ) ) ) )
7877fveq2d 5545 . 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 32403 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 10, 11, 38, 45, 47, 5, 48, 49, 24, 9, 56, 59, 16, 64, 67, 52mapdindp 32483 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
811, 2, 3, 10, 11, 38, 45, 47, 5, 56, 59, 9, 16, 64, 67, 51mapdncol 32482 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
821, 2, 3, 10, 11, 38, 45, 47, 5, 56, 59, 44, 42, 7mapdn0 32481 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 10, 11, 38, 45, 47, 5, 64, 67, 44, 42, 14mapdn0 32481 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
8445, 46, 42, 39, 47, 79, 48, 80, 81, 82, 83, 71baerlem5b 32527 . 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 2338 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } ) )  =  ( J `  { ( G  .+b  E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    i^i cin 3164   ifcif 3578   {csn 3653   {cpr 3654   <.cotp 3657    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   Basecbs 13164   +g cplusg 13224   0gc0g 13416   -gcsg 14381   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   HLchlt 30162   LHypclh 30795   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436
This theorem is referenced by:  mapdh6aN  32547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lcv 29831  df-lfl 29870  df-lkr 29898  df-ldual 29936  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tgrp 31554  df-tendo 31566  df-edring 31568  df-dveca 31814  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041  df-doch 32160  df-djh 32207  df-lcdual 32399  df-mapd 32437
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