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Theorem mapdheq4lem 30610
Description: Lemma for mapdheq4 30611. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
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.  }
) )
mapdhe4.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdhe.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
mapdh.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
mapdh.yz  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
mapdh.eg  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
mapdh.ee  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
Assertion
Ref Expression
mapdheq4lem  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( J `  { ( G R 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
Allowed substitution hints:    ph( x)    C( x)    Q( h)    U( x)    E( x)    H( x, h)    I( x, h)    K( x, h)    V( x, h)    W( x, h)

Proof of Theorem mapdheq4lem
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 2253 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 29989 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe4.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 mapdh.v . . . . . . 7  |-  V  =  ( Base `  U
)
11 mapdh.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 mapdhe.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 mapdhcl.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 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
2610, 25lmodvsubcl 15505 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
276, 24, 9, 26syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
2810, 4, 11lspsncl 15569 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
296, 27, 28syl2anc 645 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
3010, 25lmodvsubcl 15505 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
316, 24, 16, 30syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
3210, 4, 11lspsncl 15569 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
336, 31, 32syl2anc 645 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
344, 19lsmcl 15671 . . . . 5  |-  ( ( U  e.  LMod  /\  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U )  /\  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )  -> 
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) )  e.  ( LSubSp `  U
) )
356, 29, 33, 34syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) )  e.  ( LSubSp `  U
) )
361, 2, 3, 4, 5, 21, 35mapdin 30541 . . 3  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) ) ) )  =  ( ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  i^i  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) ) ) )
37 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
38 eqid 2253 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
391, 2, 3, 4, 19, 37, 38, 5, 13, 18mapdlsm 30543 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) ) )
40 mapdh.eg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
41 mapdh.q . . . . . . . . 9  |-  Q  =  ( 0g `  C
)
42 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 ) } ) ) ) ) )
43 mapdhc.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
44 mapdh.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
45 mapdh.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
46 mapdh.j . . . . . . . . 9  |-  J  =  ( LSpan `  C )
47 mapdhc.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
48 mapdh.mn . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
491, 3, 5dvhlvec 29988 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
50 mapdh.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
51 mapdh.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5210, 43, 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 } ) )
5441, 42, 1, 2, 3, 10, 25, 43, 11, 37, 44, 45, 46, 5, 47, 48, 22, 9, 53mapdhcl 30606 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5540, 54eqeltrrd 2328 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
5641, 42, 1, 2, 3, 10, 25, 43, 11, 37, 44, 45, 46, 5, 47, 48, 22, 7, 55, 53mapdheq 30607 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5740, 56mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
5857simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { G } ) )
59 mapdh.ee . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
6010, 43, 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 } ) )
6241, 42, 1, 2, 3, 10, 25, 43, 11, 37, 44, 45, 46, 5, 47, 48, 22, 16, 61mapdhcl 30606 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6359, 62eqeltrrd 2328 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
6441, 42, 1, 2, 3, 10, 25, 43, 11, 37, 44, 45, 46, 5, 47, 48, 22, 14, 63, 61mapdheq 30607 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6559, 64mpbid 203 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6665simpld 447 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6758, 66oveq12d 5728 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6839, 67eqtrd 2285 . . . 4  |-  ( ph  ->  ( M `  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
691, 2, 3, 4, 19, 37, 38, 5, 29, 33mapdlsm 30543 . . . . 5  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { ( X  .-  Z ) } ) ) ) )
7057simprd 451 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( F R G ) } ) )
7165simprd 451 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `  { ( F R E ) } ) )
7270, 71oveq12d 5728 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { ( X 
.-  Z ) } ) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) )
7369, 72eqtrd 2285 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) )
7468, 73ineq12d 3279 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) ) ) )  =  ( ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) ) )
7536, 74eqtrd 2285 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  {
( X  .-  Z
) } ) ) ) )  =  ( ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) ) )
7610, 25, 43, 19, 11, 49, 24, 51, 50, 7, 14baerlem3 30592 . . 3  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { ( X  .-  Z ) } ) ) ) )
7776fveq2d 5381 . 2  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( M `  ( ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) ) ) )
78 eqid 2253 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
791, 37, 5lcdlvec 30470 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 10, 11, 37, 44, 46, 5, 47, 48, 24, 9, 55, 58, 16, 63, 66, 51mapdindp 30550 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
811, 2, 3, 10, 11, 37, 44, 46, 5, 55, 58, 9, 16, 63, 66, 50mapdncol 30549 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
821, 2, 3, 10, 11, 37, 44, 46, 5, 55, 58, 43, 78, 7mapdn0 30548 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { ( 0g
`  C ) } ) )
831, 2, 3, 10, 11, 37, 44, 46, 5, 63, 66, 43, 78, 14mapdn0 30548 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { ( 0g
`  C ) } ) )
8444, 45, 78, 38, 46, 79, 47, 80, 81, 82, 83baerlem3 30592 . 2  |-  ( ph  ->  ( J `  {
( G R E ) } )  =  ( ( ( J `
 { G }
) ( LSSum `  C
) ( J `  { E } ) )  i^i  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) ) )
8575, 77, 843eqtr4d 2295 1  |-  ( ph  ->  ( M `  ( N `  { ( Y  .-  Z ) } ) )  =  ( J `  { ( G R E ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   _Vcvv 2727    \ cdif 3075    i^i cin 3077   ifcif 3470   {csn 3544   {cpr 3545   <.cotp 3548    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   1stc1st 5972   2ndc2nd 5973   iota_crio 6181   Basecbs 13022   0gc0g 13274   -gcsg 14200   LSSumclsm 14780   LModclmod 15462   LSubSpclss 15524   LSpanclspn 15563   HLchlt 28229   LHypclh 28862   DVecHcdvh 29957  LCDualclcd 30465  mapdcmpd 30503
This theorem is referenced by:  mapdheq4  30611
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 27855  df-lshyp 27856  df-lcv 27898  df-lfl 27937  df-lkr 27965  df-ldual 28003  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-tgrp 29621  df-tendo 29633  df-edring 29635  df-dveca 29881  df-disoa 29908  df-dvech 29958  df-dib 30018  df-dic 30052  df-dih 30108  df-doch 30227  df-djh 30274  df-lcdual 30466  df-mapd 30504
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