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Theorem mapdh6dN 31854
Description: Lemmma for mapdh6N 31862. (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 31707 . . . . 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.  }
) )
2019eldifad 3275 . . . . . 6  |-  ( ph  ->  w  e.  V )
211, 8, 3dvhlvec 31224 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
2218eldifad 3275 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
23 mapdh6d.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2423eldifad 3275 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
25 mapdh6d.wn . . . . . . . . 9  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
269, 12, 21, 20, 22, 24, 25lspindpi 16131 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { Y } ) ) )
2726simpld 446 . . . . . . 7  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { X } ) )
2827necomd 2633 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { w } ) )
295, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 3, 16, 17, 18, 20, 28mapdhcl 31842 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  w >. )  e.  D )
30 mapdh.a . . . . . 6  |-  .+b  =  ( +g  `  C )
3113, 30, 5lmod0vrid 15908 . . . . 5  |-  ( ( C  e.  LMod  /\  (
I `  <. X ,  F ,  w >. )  e.  D )  -> 
( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
324, 29, 31syl2anc 643 . . . 4  |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
3332adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  Q )  =  ( I `  <. X ,  F ,  w >. ) )
34 oteq3 3937 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  <. X ,  F ,  ( Y  .+  Z ) >.  =  <. X ,  F ,  .0.  >.
)
3534fveq2d 5672 . . . . 5  |-  ( ( Y  .+  Z )  =  .0.  ->  (
I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( I `  <. X ,  F ,  .0.  >. ) )
365, 6, 11, 18, 16mapdhval0 31840 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
3735, 36sylan9eqr 2441 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  Q )
3837oveq2d 6036 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  ( I `  <. X ,  F , 
( Y  .+  Z
) >. ) )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
) )
39 oveq2 6028 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  (
w  .+  ( Y  .+  Z ) )  =  ( w  .+  .0.  ) )
401, 8, 3dvhlmod 31225 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
41 mapdh.p . . . . . . . 8  |-  .+  =  ( +g  `  U )
429, 41, 11lmod0vrid 15908 . . . . . . 7  |-  ( ( U  e.  LMod  /\  w  e.  V )  ->  (
w  .+  .0.  )  =  w )
4340, 20, 42syl2anc 643 . . . . . 6  |-  ( ph  ->  ( w  .+  .0.  )  =  w )
4439, 43sylan9eqr 2441 . . . . 5  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( w  .+  ( Y  .+  Z
) )  =  w )
4544oteq3d 3940 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
4645fveq2d 5672 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( I `  <. X ,  F ,  w >. ) )
4733, 38, 463eqtr4rd 2430 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
483adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
4916adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  F  e.  D
)
5017adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
5118adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
5219adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  w  e.  ( V  \  {  .0.  } ) )
53 mapdh6d.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5453eldifad 3275 . . . . . 6  |-  ( ph  ->  Z  e.  V )
559, 41lmodvacl 15891 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
5640, 24, 54, 55syl3anc 1184 . . . . 5  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
5756anim1i 552 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( ( Y 
.+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
58 eldifsn 3870 . . . 4  |-  ( ( Y  .+  Z )  e.  ( V  \  {  .0.  } )  <->  ( ( Y  .+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
5957, 58sylibr 204 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( Y  .+  Z )  e.  ( V  \  {  .0.  } ) )
60 mapdh6d.yz . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
61 mapdh6d.xn . . . . . . . . 9  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
629, 12, 21, 22, 24, 54, 61lspindpi 16131 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
6362simpld 446 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
649, 41, 11, 12, 21, 18, 23, 53, 19, 60, 63, 25mapdindp1 31835 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
659, 41, 11, 12, 21, 18, 23, 53, 19, 60, 63, 25mapdindp2 31836 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  ( Y  .+  Z ) } ) )
669, 11, 12, 21, 18, 56, 20, 64, 65lspindp1 16132 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } )  /\  -.  X  e.  ( N `  {
w ,  ( Y 
.+  Z ) } ) ) )
6766simprd 450 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6867adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6926simprd 450 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { Y } ) )
709, 11, 12, 21, 19, 24, 69lspsnne1 16116 . . . . . . . 8  |-  ( ph  ->  -.  w  e.  ( N `  { Y } ) )
71 eqid 2387 . . . . . . . . . 10  |-  ( LSSum `  U )  =  (
LSSum `  U )
729, 12, 71, 40, 24, 54lsmpr 16088 . . . . . . . . 9  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )
7360oveq2d 6036 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )
74 eqid 2387 . . . . . . . . . . . . 13  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
759, 74, 12lspsncl 15980 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7640, 24, 75syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7774lsssubg 15960 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( N `  { Y } )  e.  (SubGrp `  U )
)
7840, 76, 77syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  U ) )
7971lsmidm 15223 . . . . . . . . . 10  |-  ( ( N `  { Y } )  e.  (SubGrp `  U )  ->  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Y }
) )  =  ( N `  { Y } ) )
8078, 79syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( N `  { Y } ) )
8172, 73, 803eqtr2d 2425 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( N `
 { Y }
) )
8270, 81neleqtrrd 2483 . . . . . . 7  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  Z } ) )
839, 41, 12, 40, 24, 54, 20, 82lspindp4 16136 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  ( Y  .+  Z ) } ) )
849, 12, 21, 20, 24, 56, 83lspindpi 16131 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { Y } )  /\  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) ) )
8584simprd 450 . . . 4  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  {
( Y  .+  Z
) } ) )
8685adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
87 eqidd 2388 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F ,  w >. )  =  ( I `  <. X ,  F ,  w >. ) )
88 eqidd 2388 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( Y  .+  Z
) >. )  =  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
)
895, 6, 1, 7, 8, 9, 10, 11, 12, 2, 13, 14, 15, 48, 49, 50, 51, 41, 30, 52, 59, 68, 86, 87, 88mapdh6aN 31850 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( w  .+  ( Y  .+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
9047, 89pm2.61dane 2628 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 359    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    \ cdif 3260   ifcif 3682   {csn 3757   {cpr 3758   <.cotp 3761    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   iota_crio 6478   Basecbs 13396   +g cplusg 13456   0gc0g 13650   -gcsg 14615  SubGrpcsubg 14865   LSSumclsm 15195   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974   HLchlt 29465   LHypclh 30098   DVecHcdvh 31193  LCDualclcd 31701  mapdcmpd 31739
This theorem is referenced by:  mapdh6gN  31857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-ot 3767  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-tpos 6415  df-undef 6479  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-0g 13654  df-mre 13738  df-mrc 13739  df-acs 13741  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-cntz 15043  df-oppg 15069  df-lsm 15197  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-dvr 15715  df-drng 15764  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lvec 16102  df-lsatoms 29091  df-lshyp 29092  df-lcv 29134  df-lfl 29173  df-lkr 29201  df-ldual 29239  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tgrp 30857  df-tendo 30869  df-edring 30871  df-dveca 31117  df-disoa 31144  df-dvech 31194  df-dib 31254  df-dic 31288  df-dih 31344  df-doch 31463  df-djh 31510  df-lcdual 31702  df-mapd 31740
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