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Theorem mapdh6lem1N 31996
Description: Lemma for mapdh6N 32010. Part (6) in [Baer] p. 47, lines 16-18. (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
mapdh6lem1N  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( 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 mapdh6lem1N
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 2285 . . . 4  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
5 mapdh.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 3, 5dvhlmod 31373 . . . . 5  |-  ( ph  ->  U  e.  LMod )
7 mapdhcl.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
8 eldifi 3300 . . . . . . . 8  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  X  e.  V )
10 mapdhe6.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3300 . . . . . . . 8  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1210, 11syl 15 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
13 mapdh.v . . . . . . . 8  |-  V  =  ( Base `  U
)
14 mapdh.s . . . . . . . 8  |-  .-  =  ( -g `  U )
1513, 14lmodvsubcl 15672 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
166, 9, 12, 15syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( X  .-  Y
)  e.  V )
17 mapdh.n . . . . . . 7  |-  N  =  ( LSpan `  U )
1813, 4, 17lspsncl 15736 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Y )  e.  V )  ->  ( N `  { ( X  .-  Y ) } )  e.  ( LSubSp `  U ) )
196, 16, 18syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Y
) } )  e.  ( LSubSp `  U )
)
20 mapdhe6.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
21 eldifi 3300 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
2220, 21syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  V )
2313, 4, 17lspsncl 15736 . . . . . 6  |-  ( ( U  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
246, 22, 23syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  U ) )
25 eqid 2285 . . . . . 6  |-  ( LSSum `  U )  =  (
LSSum `  U )
264, 25lsmcl 15838 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  e.  ( LSubSp `  U )
)
2813, 14lmodvsubcl 15672 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V  /\  Z  e.  V )  ->  ( X  .-  Z )  e.  V )
296, 9, 22, 28syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( X  .-  Z
)  e.  V )
3013, 4, 17lspsncl 15736 . . . . . 6  |-  ( ( U  e.  LMod  /\  ( X  .-  Z )  e.  V )  ->  ( N `  { ( X  .-  Z ) } )  e.  ( LSubSp `  U ) )
316, 29, 30syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  {
( X  .-  Z
) } )  e.  ( LSubSp `  U )
)
3213, 4, 17lspsncl 15736 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
336, 12, 32syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
344, 25lsmcl 15838 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) )  e.  ( LSubSp `  U )
)
361, 2, 3, 4, 5, 27, 35mapdin 31925 . . 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 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
38 eqid 2285 . . . . . 6  |-  ( LSSum `  C )  =  (
LSSum `  C )
391, 2, 3, 4, 25, 37, 38, 5, 19, 24mapdlsm 31927 . . . . 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 31927 . . . . 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 mapdh6.fg . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
43 mapdh.q . . . . . . . . 9  |-  Q  =  ( 0g `  C
)
44 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 ) } ) ) ) ) )
45 mapdhc.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
46 mapdh.d . . . . . . . . 9  |-  D  =  ( Base `  C
)
47 mapdh.r . . . . . . . . 9  |-  R  =  ( -g `  C
)
48 mapdh.j . . . . . . . . 9  |-  J  =  ( LSpan `  C )
49 mapdhc.f . . . . . . . . 9  |-  ( ph  ->  F  e.  D )
50 mapdh.mn . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
511, 3, 5dvhlvec 31372 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  LVec )
52 mapdh6.yz . . . . . . . . . . . . 13  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
53 mapdhe6.xn . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
5413, 45, 17, 51, 12, 20, 9, 52, 53lspindp2 15890 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  -.  Z  e.  ( N `  { X ,  Y } ) ) )
5554simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
5643, 44, 1, 2, 3, 13, 14, 45, 17, 37, 46, 47, 48, 5, 49, 50, 7, 12, 55mapdhcl 31990 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
5742, 56eqeltrrd 2360 . . . . . . . . 9  |-  ( ph  ->  G  e.  D )
5843, 44, 1, 2, 3, 13, 14, 45, 17, 37, 46, 47, 48, 5, 49, 50, 7, 10, 57, 55mapdheq 31991 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) ) )
5942, 58mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R G ) } ) ) )
6059simprd 449 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( F R G ) } ) )
61 mapdh6.fe . . . . . . . 8  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )
6213, 45, 17, 51, 10, 22, 9, 52, 53lspindp1 15888 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Z } )  /\  -.  Y  e.  ( N `  { X ,  Z } ) ) )
6362simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
6443, 44, 1, 2, 3, 13, 14, 45, 17, 37, 46, 47, 48, 5, 49, 50, 7, 22, 63mapdhcl 31990 . . . . . . . . . 10  |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  e.  D )
6561, 64eqeltrrd 2360 . . . . . . . . 9  |-  ( ph  ->  E  e.  D )
6643, 44, 1, 2, 3, 13, 14, 45, 17, 37, 46, 47, 48, 5, 49, 50, 7, 20, 65, 63mapdheq 31991 . . . . . . . 8  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Z >. )  =  E  <-> 
( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) ) )
6761, 66mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { Z } ) )  =  ( J `  { E } )  /\  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `
 { ( F R E ) } ) ) )
6867simpld 445 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Z } ) )  =  ( J `  { E } ) )
6960, 68oveq12d 5878 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Y ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Z }
) ) )  =  ( ( J `  { ( F R G ) } ) ( LSSum `  C )
( J `  { E } ) ) )
7067simprd 449 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Z ) } ) )  =  ( J `  { ( F R E ) } ) )
7159simpld 445 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { G } ) )
7270, 71oveq12d 5878 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( X  .-  Z ) } ) ) (
LSSum `  C ) ( M `  ( N `
 { Y }
) ) )  =  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) )
7369, 72ineq12d 3373 . . . 4  |-  ( ph  ->  ( ( ( M `
 ( N `  { ( X  .-  Y ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Z } ) ) )  i^i  ( ( M `
 ( N `  { ( X  .-  Z ) } ) ) ( LSSum `  C
) ( M `  ( N `  { Y } ) ) ) )  =  ( ( ( J `  {
( F R G ) } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) ) )
7441, 73eqtrd 2317 . . 3  |-  ( ph  ->  ( ( M `  ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) ) )  i^i  ( M `  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( J `  {
( F R G ) } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) ) )
7536, 74eqtrd 2317 . 2  |-  ( ph  ->  ( M `  (
( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )  =  ( ( ( J `  {
( F R G ) } ) (
LSSum `  C ) ( J `  { E } ) )  i^i  ( ( J `  { ( F R E ) } ) ( LSSum `  C )
( J `  { G } ) ) ) )
76 mapdh.p . . . 4  |-  .+  =  ( +g  `  U )
7713, 14, 45, 25, 17, 51, 9, 53, 52, 10, 20, 76baerlem5a 31977 . . 3  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) ( LSSum `  U )
( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) )
7877fveq2d 5531 . 2  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( M `  ( ( ( N `  {
( X  .-  Y
) } ) (
LSSum `  U ) ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .-  Z ) } ) ( LSSum `  U )
( N `  { Y } ) ) ) ) )
791, 37, 5lcdlvec 31854 . . 3  |-  ( ph  ->  C  e.  LVec )
801, 2, 3, 13, 17, 37, 46, 48, 5, 49, 50, 9, 12, 57, 71, 22, 65, 68, 53mapdindp 31934 . . 3  |-  ( ph  ->  -.  F  e.  ( J `  { G ,  E } ) )
811, 2, 3, 13, 17, 37, 46, 48, 5, 57, 71, 12, 22, 65, 68, 52mapdncol 31933 . . 3  |-  ( ph  ->  ( J `  { G } )  =/=  ( J `  { E } ) )
821, 2, 3, 13, 17, 37, 46, 48, 5, 57, 71, 45, 43, 10mapdn0 31932 . . 3  |-  ( ph  ->  G  e.  ( D 
\  { Q }
) )
831, 2, 3, 13, 17, 37, 46, 48, 5, 65, 68, 45, 43, 20mapdn0 31932 . . 3  |-  ( ph  ->  E  e.  ( D 
\  { Q }
) )
84 mapdh.a . . 3  |-  .+b  =  ( +g  `  C )
8546, 47, 43, 38, 48, 79, 49, 80, 81, 82, 83, 84baerlem5a 31977 . 2  |-  ( ph  ->  ( J `  {
( F R ( G  .+b  E )
) } )  =  ( ( ( J `
 { ( F R G ) } ) ( LSSum `  C
) ( J `  { E } ) )  i^i  ( ( J `
 { ( F R E ) } ) ( LSSum `  C
) ( J `  { G } ) ) ) )
8675, 78, 853eqtr4d 2327 1  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( J `  { ( F R ( G 
.+b  E ) ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   _Vcvv 2790    \ cdif 3151    i^i cin 3153   ifcif 3567   {csn 3642   {cpr 3643   <.cotp 3646    e. cmpt 4079   ` cfv 5257  (class class class)co 5860   1stc1st 6122   2ndc2nd 6123   iota_crio 6299   Basecbs 13150   +g cplusg 13210   0gc0g 13402   -gcsg 14367   LSSumclsm 14947   LModclmod 15629   LSubSpclss 15691   LSpanclspn 15730   HLchlt 29613   LHypclh 30246   DVecHcdvh 31341  LCDualclcd 31849  mapdcmpd 31887
This theorem is referenced by:  mapdh6lem2N  31997  mapdh6aN  31998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  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 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mre 13490  df-mrc 13491  df-acs 13493  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-cntz 14795  df-oppg 14821  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-lsatoms 29239  df-lshyp 29240  df-lcv 29282  df-lfl 29321  df-lkr 29349  df-ldual 29387  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tgrp 31005  df-tendo 31017  df-edring 31019  df-dveca 31265  df-disoa 31292  df-dvech 31342  df-dib 31402  df-dic 31436  df-dih 31492  df-doch 31611  df-djh 31658  df-lcdual 31850  df-mapd 31888
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