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Theorem mapdheq4lem 31088
Description: Lemma for mapdheq4 31089. 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 2258 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 30467 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhe4.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3273 . . . . . . 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 15696 . . . . . 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 3273 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
1614, 15syl 17 . . . . . 6  |-  ( ph  ->  Z  e.  V )
1710, 4, 11lspsncl 15696 . . . . . 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 2258 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
204, 19lsmcl 15798 . . . . 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 3273 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2422, 23syl 17 . . . . . . 7  |-  ( ph  ->  X  e.  V )
25 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
2610, 25lmodvsubcl 15632 . . . . . . 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 15696 . . . . . 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 15632 . . . . . . 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 15696 . . . . . 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 15798 . . . . 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 31019 . . 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 2258 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
391, 2, 3, 4, 19, 37, 38, 5, 13, 18mapdlsm 31021 . . . . 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 30466 . . . . . . . . . . . . 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 15850 . . . . . . . . . . . 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 31084 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5540, 54eqeltrrd 2333 . . . . . . . . 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 31085 . . . . . . . 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 15848 . . . . . . . . . . . 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 31084 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6359, 62eqeltrrd 2333 . . . . . . . . 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 31085 . . . . . . . 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 5810 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { G } ) (
LSSum `  C ) ( J `  { E } ) ) )
6839, 67eqtrd 2290 . . . 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 31021 . . . . 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 5810 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { ( X 
.-  Z ) } ) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  {
( F R E ) } ) ) )
7369, 72eqtrd 2290 . . . 4  |-  ( ph  ->  ( M `  (
( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { ( X  .-  Z ) } ) ) )  =  ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { ( F R E ) } ) ) )
7468, 73ineq12d 3346 . . 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 2290 . 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 31070 . . 3  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
) ( LSSum `  U
) ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  Y ) } ) ( LSSum `  U
) ( N `  { ( X  .-  Z ) } ) ) ) )
7776fveq2d 5462 . 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 2258 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
791, 37, 5lcdlvec 30948 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 10, 11, 37, 44, 46, 5, 47, 48, 24, 9, 55, 58, 16, 63, 66, 51mapdindp 31028 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
811, 2, 3, 10, 11, 37, 44, 46, 5, 55, 58, 9, 16, 63, 66, 50mapdncol 31027 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
821, 2, 3, 10, 11, 37, 44, 46, 5, 55, 58, 43, 78, 7mapdn0 31026 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { ( 0g
`  C ) } ) )
831, 2, 3, 10, 11, 37, 44, 46, 5, 63, 66, 43, 78, 14mapdn0 31026 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { ( 0g
`  C ) } ) )
8444, 45, 78, 38, 46, 79, 47, 80, 81, 82, 83baerlem3 31070 . 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 2300 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 2421   _Vcvv 2763    \ cdif 3124    i^i cin 3126   ifcif 3539   {csn 3614   {cpr 3615   <.cotp 3618    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   1stc1st 6054   2ndc2nd 6055   iota_crio 6263   Basecbs 13110   0gc0g 13362   -gcsg 14327   LSSumclsm 14907   LModclmod 15589   LSubSpclss 15651   LSpanclspn 15690   HLchlt 28707   LHypclh 29340   DVecHcdvh 30435  LCDualclcd 30943  mapdcmpd 30981
This theorem is referenced by:  mapdheq4  31089
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9933  df-z 9992  df-uz 10198  df-fz 10749  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-sca 13186  df-vsca 13187  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lshyp 28334  df-lcv 28376  df-lfl 28415  df-lkr 28443  df-ldual 28481  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tgrp 30099  df-tendo 30111  df-edring 30113  df-dveca 30359  df-disoa 30386  df-dvech 30436  df-dib 30496  df-dic 30530  df-dih 30586  df-doch 30705  df-djh 30752  df-lcdual 30944  df-mapd 30982
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