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Theorem mapdh6dN 32551
Description: Lemmma for mapdh6N 32559. (Contributed by NM, 1-May-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 )
mapdh6d.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
mapdh6d.yz  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
mapdh6d.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdh6d.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
mapdh6d.w  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
mapdh6d.wn  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
Assertion
Ref Expression
mapdh6dN  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
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    w, h    h, Z, x    .+b , h    h, I, x    .+ , h, x   
x, w
Allowed substitution hints:    ph( x, w)    C( x, w)    D( w)    .+ ( w)    .+b ( x, w)    Q( w, h)    R( w)    U( x, w)    F( w)    H( x, w, h)    I( w)    J( w)    K( x, w, h)    M( w)    .- ( w)    N( w)    V( x, w, h)    W( x, w, h)    X( w)    Y( w)    .0. ( w)    Z( w)

Proof of Theorem mapdh6dN
StepHypRef Expression
1 mapdh.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 mapdh.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
3 mapdh.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 32404 . . . . 5  |-  ( ph  ->  C  e.  LMod )
5 mapdh.q . . . . . 6  |-  Q  =  ( 0g `  C
)
6 mapdh.i . . . . . 6  |-  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 ) } ) ) ) ) )
7 mapdh.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
8 mapdh.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
9 mapdh.v . . . . . 6  |-  V  =  ( Base `  U
)
10 mapdh.s . . . . . 6  |-  .-  =  ( -g `  U )
11 mapdhc.o . . . . . 6  |-  .0.  =  ( 0g `  U )
12 mapdh.n . . . . . 6  |-  N  =  ( LSpan `  U )
13 mapdh.d . . . . . 6  |-  D  =  ( Base `  C
)
14 mapdh.r . . . . . 6  |-  R  =  ( -g `  C
)
15 mapdh.j . . . . . 6  |-  J  =  ( LSpan `  C )
16 mapdhc.f . . . . . 6  |-  ( ph  ->  F  e.  D )
17 mapdh.mn . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
18 mapdhcl.x . . . . . 6  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
19 mapdh6d.w . . . . . . 7  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
20 eldifi 3311 . . . . . . 7  |-  ( w  e.  ( V  \  {  .0.  } )  ->  w  e.  V )
2119, 20syl 15 . . . . . 6  |-  ( ph  ->  w  e.  V )
221, 8, 3dvhlvec 31921 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
23 eldifi 3311 . . . . . . . . . 10  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2418, 23syl 15 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
25 mapdh6d.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
26 eldifi 3311 . . . . . . . . . 10  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
2725, 26syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
28 mapdh6d.wn . . . . . . . . 9  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
299, 12, 22, 21, 24, 27, 28lspindpi 15901 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { Y } ) ) )
3029simpld 445 . . . . . . 7  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { X } ) )
3130necomd 2542 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { w } ) )
325, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 3, 16, 17, 18, 21, 31mapdhcl 32539 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  w >. )  e.  D )
33 mapdh.a . . . . . 6  |-  .+b  =  ( +g  `  C )
3413, 33, 5lmod0vrid 15677 . . . . 5  |-  ( ( C  e.  LMod  /\  (
I `  <. X ,  F ,  w >. )  e.  D )  -> 
( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
354, 32, 34syl2anc 642 . . . 4  |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
3635adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  Q )  =  ( I `  <. X ,  F ,  w >. ) )
37 oteq3 3823 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  <. X ,  F ,  ( Y  .+  Z ) >.  =  <. X ,  F ,  .0.  >.
)
3837fveq2d 5545 . . . . 5  |-  ( ( Y  .+  Z )  =  .0.  ->  (
I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( I `  <. X ,  F ,  .0.  >. ) )
395, 6, 11, 18, 16mapdhval0 32537 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
4038, 39sylan9eqr 2350 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  Q )
4140oveq2d 5890 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  ( I `  <. X ,  F , 
( Y  .+  Z
) >. ) )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
) )
42 oveq2 5882 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  (
w  .+  ( Y  .+  Z ) )  =  ( w  .+  .0.  ) )
431, 8, 3dvhlmod 31922 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
44 mapdh.p . . . . . . . 8  |-  .+  =  ( +g  `  U )
459, 44, 11lmod0vrid 15677 . . . . . . 7  |-  ( ( U  e.  LMod  /\  w  e.  V )  ->  (
w  .+  .0.  )  =  w )
4643, 21, 45syl2anc 642 . . . . . 6  |-  ( ph  ->  ( w  .+  .0.  )  =  w )
4742, 46sylan9eqr 2350 . . . . 5  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( w  .+  ( Y  .+  Z
) )  =  w )
48 oteq3 3823 . . . . 5  |-  ( ( w  .+  ( Y 
.+  Z ) )  =  w  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
4947, 48syl 15 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
5049fveq2d 5545 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( I `  <. X ,  F ,  w >. ) )
5136, 41, 503eqtr4rd 2339 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
523adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
5316adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  F  e.  D
)
5417adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
5518adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
5619adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  w  e.  ( V  \  {  .0.  } ) )
57 mapdh6d.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
58 eldifi 3311 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
5957, 58syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  V )
609, 44lmodvacl 15657 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
6143, 27, 59, 60syl3anc 1182 . . . . 5  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
6261anim1i 551 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( ( Y 
.+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
63 eldifsn 3762 . . . 4  |-  ( ( Y  .+  Z )  e.  ( V  \  {  .0.  } )  <->  ( ( Y  .+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
6462, 63sylibr 203 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( Y  .+  Z )  e.  ( V  \  {  .0.  } ) )
65 mapdh6d.yz . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
66 mapdh6d.xn . . . . . . . . 9  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
679, 12, 22, 24, 27, 59, 66lspindpi 15901 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
6867simpld 445 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
699, 44, 11, 12, 22, 18, 25, 57, 19, 65, 68, 28mapdindp1 32532 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
709, 44, 11, 12, 22, 18, 25, 57, 19, 65, 68, 28mapdindp2 32533 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  ( Y  .+  Z ) } ) )
719, 11, 12, 22, 18, 61, 21, 69, 70lspindp1 15902 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } )  /\  -.  X  e.  ( N `  {
w ,  ( Y 
.+  Z ) } ) ) )
7271simprd 449 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
7372adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
7429simprd 449 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { Y } ) )
759, 11, 12, 22, 19, 27, 74lspsnne1 15886 . . . . . . . 8  |-  ( ph  ->  -.  w  e.  ( N `  { Y } ) )
76 eqid 2296 . . . . . . . . . . 11  |-  ( LSSum `  U )  =  (
LSSum `  U )
779, 12, 76, 43, 27, 59lsmpr 15858 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )
7865oveq2d 5890 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )
79 eqid 2296 . . . . . . . . . . . . . 14  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
809, 79, 12lspsncl 15750 . . . . . . . . . . . . 13  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
8143, 27, 80syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
8279lsssubg 15730 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( N `  { Y } )  e.  (SubGrp `  U )
)
8343, 81, 82syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  U ) )
8476lsmidm 14989 . . . . . . . . . . 11  |-  ( ( N `  { Y } )  e.  (SubGrp `  U )  ->  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Y }
) )  =  ( N `  { Y } ) )
8583, 84syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( N `  { Y } ) )
8677, 78, 853eqtr2d 2334 . . . . . . . . 9  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( N `
 { Y }
) )
8786eleq2d 2363 . . . . . . . 8  |-  ( ph  ->  ( w  e.  ( N `  { Y ,  Z } )  <->  w  e.  ( N `  { Y } ) ) )
8875, 87mtbird 292 . . . . . . 7  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  Z } ) )
899, 44, 12, 43, 27, 59, 21, 88lspindp4 15906 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  ( Y  .+  Z ) } ) )
909, 12, 22, 21, 27, 61, 89lspindpi 15901 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { Y } )  /\  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) ) )
9190simprd 449 . . . 4  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  {
( Y  .+  Z
) } ) )
9291adantr 451 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
93 eqidd 2297 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F ,  w >. )  =  ( I `  <. X ,  F ,  w >. ) )
94 eqidd 2297 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( Y  .+  Z
) >. )  =  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
)
955, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 52, 53, 54, 55, 44, 33, 56, 64, 73, 92, 93, 94mapdh6aN 32547 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( w  .+  ( Y  .+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
9651, 95pm2.61dane 2537 1  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
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   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  SubGrpcsubg 14631   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   HLchlt 30162   LHypclh 30795   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436
This theorem is referenced by:  mapdh6gN  32554
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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