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Theorem lcfrlem16 31817
Description: Lemma for lcfr 31844. (Contributed by NM, 8-Mar-2015.)
Hypotheses
Ref Expression
lcf1o.h  |-  H  =  ( LHyp `  K
)
lcf1o.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcf1o.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcf1o.v  |-  V  =  ( Base `  U
)
lcf1o.a  |-  .+  =  ( +g  `  U )
lcf1o.t  |-  .x.  =  ( .s `  U )
lcf1o.s  |-  S  =  (Scalar `  U )
lcf1o.r  |-  R  =  ( Base `  S
)
lcf1o.z  |-  .0.  =  ( 0g `  U )
lcf1o.f  |-  F  =  (LFnl `  U )
lcf1o.l  |-  L  =  (LKer `  U )
lcf1o.d  |-  D  =  (LDual `  U )
lcf1o.q  |-  Q  =  ( 0g `  D
)
lcf1o.c  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
lcf1o.j  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
lcflo.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfrlem16.p  |-  P  =  ( LSubSp `  D )
lcfrlem16.g  |-  ( ph  ->  G  e.  P )
lcfrlem16.gs  |-  ( ph  ->  G  C_  C )
lcfrlem16.m  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
lcfrlem16.x  |-  ( ph  ->  X  e.  ( E 
\  {  .0.  }
) )
Assertion
Ref Expression
lcfrlem16  |-  ( ph  ->  ( J `  X
)  e.  G )
Distinct variable groups:    x, w,  ._|_    x,  .0.    x, v, V    x,  .x.    v, k, w, x, X    x,  .+    x, R   
f, k, v, w, 
.+    f, F, k    g,
k, G    f, g, J, k    f, L, k    ._|_ , f, k, v    R, f, k, v    S, k    .x. , f, k, v, w    U, k    f, V, g, x    f, X    v,
g, w, x, X    ph, g, k
Allowed substitution hints:    ph( x, w, v, f)    C( x, w, v, f, g, k)    D( x, w, v, f, g, k)    P( x, w, v, f, g, k)    .+ ( g)    Q( x, w, v, f, g, k)    R( w, g)    S( x, w, v, f, g)    .x. ( g)    U( x, w, v, f, g)    E( x, w, v, f, g, k)    F( x, w, v, g)    G( x, w, v, f)    H( x, w, v, f, g, k)    J( x, w, v)    K( x, w, v, f, g, k)    L( x, w, v, g)    ._|_ ( g)    V( w, k)    W( x, w, v, f, g, k)    .0. ( w, v, f, g, k)

Proof of Theorem lcfrlem16
StepHypRef Expression
1 lcfrlem16.x . . . . 5  |-  ( ph  ->  X  e.  ( E 
\  {  .0.  }
) )
2 eldifi 3374 . . . . 5  |-  ( X  e.  ( E  \  {  .0.  } )  ->  X  e.  E )
31, 2syl 15 . . . 4  |-  ( ph  ->  X  e.  E )
4 lcfrlem16.m . . . 4  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
53, 4syl6eleq 2448 . . 3  |-  ( ph  ->  X  e.  U_ g  e.  G  (  ._|_  `  ( L `  g
) ) )
6 eliun 3990 . . 3  |-  ( X  e.  U_ g  e.  G  (  ._|_  `  ( L `  g )
)  <->  E. g  e.  G  X  e.  (  ._|_  `  ( L `  g
) ) )
75, 6sylib 188 . 2  |-  ( ph  ->  E. g  e.  G  X  e.  (  ._|_  `  ( L `  g
) ) )
8 lcf1o.s . . . . 5  |-  S  =  (Scalar `  U )
9 lcf1o.r . . . . 5  |-  R  =  ( Base `  S
)
10 lcf1o.f . . . . 5  |-  F  =  (LFnl `  U )
11 lcf1o.l . . . . 5  |-  L  =  (LKer `  U )
12 lcf1o.d . . . . 5  |-  D  =  (LDual `  U )
13 eqid 2358 . . . . 5  |-  ( .s
`  D )  =  ( .s `  D
)
14 lcf1o.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
15 lcf1o.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
16 lcflo.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
1714, 15, 16dvhlvec 31368 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
18173ad2ant1 976 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  U  e.  LVec )
19 lcfrlem16.g . . . . . . . 8  |-  ( ph  ->  G  e.  P )
20 eqid 2358 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
21 lcfrlem16.p . . . . . . . . 9  |-  P  =  ( LSubSp `  D )
2220, 21lssel 15794 . . . . . . . 8  |-  ( ( G  e.  P  /\  g  e.  G )  ->  g  e.  ( Base `  D ) )
2319, 22sylan 457 . . . . . . 7  |-  ( (
ph  /\  g  e.  G )  ->  g  e.  ( Base `  D
) )
2414, 15, 16dvhlmod 31369 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
2510, 12, 20, 24ldualvbase 29385 . . . . . . . 8  |-  ( ph  ->  ( Base `  D
)  =  F )
2625adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  e.  G )  ->  ( Base `  D )  =  F )
2723, 26eleqtrd 2434 . . . . . 6  |-  ( (
ph  /\  g  e.  G )  ->  g  e.  F )
28273adant3 975 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  F
)
29 lcf1o.o . . . . . . 7  |-  ._|_  =  ( ( ocH `  K
) `  W )
30 lcf1o.v . . . . . . 7  |-  V  =  ( Base `  U
)
31 lcf1o.a . . . . . . 7  |-  .+  =  ( +g  `  U )
32 lcf1o.t . . . . . . 7  |-  .x.  =  ( .s `  U )
33 lcf1o.z . . . . . . 7  |-  .0.  =  ( 0g `  U )
34 lcf1o.q . . . . . . 7  |-  Q  =  ( 0g `  D
)
35 lcf1o.c . . . . . . 7  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
36 lcf1o.j . . . . . . 7  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
3716adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
3824adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  g  e.  G )  ->  U  e.  LMod )
3930, 10, 11, 38, 27lkrssv 29355 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  G )  ->  ( L `  g )  C_  V )
4014, 15, 30, 29dochssv 31614 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  g )  C_  V
)  ->  (  ._|_  `  ( L `  g
) )  C_  V
)
4137, 39, 40syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  G )  ->  (  ._|_  `  ( L `  g ) )  C_  V )
4241ralrimiva 2702 . . . . . . . . . . 11  |-  ( ph  ->  A. g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
43 iunss 4024 . . . . . . . . . . 11  |-  ( U_ g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V 
<-> 
A. g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
4442, 43sylibr 203 . . . . . . . . . 10  |-  ( ph  ->  U_ g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
454, 44syl5eqss 3298 . . . . . . . . 9  |-  ( ph  ->  E  C_  V )
46 ssdif 3387 . . . . . . . . 9  |-  ( E 
C_  V  ->  ( E  \  {  .0.  }
)  C_  ( V  \  {  .0.  } ) )
4745, 46syl 15 . . . . . . . 8  |-  ( ph  ->  ( E  \  {  .0.  } )  C_  ( V  \  {  .0.  }
) )
4847, 1sseldd 3257 . . . . . . 7  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
4914, 29, 15, 30, 31, 32, 8, 9, 33, 10, 11, 12, 34, 35, 36, 16, 48lcfrlem10 31811 . . . . . 6  |-  ( ph  ->  ( J `  X
)  e.  F )
50493ad2ant1 976 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( J `  X )  e.  F
)
51 eqid 2358 . . . . . . 7  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
52163ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
53 simp3 957 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( 
._|_  `  ( L `  g ) ) )
54 eldifsni 3826 . . . . . . . . . . 11  |-  ( X  e.  ( E  \  {  .0.  } )  ->  X  =/=  .0.  )
551, 54syl 15 . . . . . . . . . 10  |-  ( ph  ->  X  =/=  .0.  )
56553ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  =/=  .0.  )
57 eldifsn 3825 . . . . . . . . 9  |-  ( X  e.  ( (  ._|_  `  ( L `  g
) )  \  {  .0.  } )  <->  ( X  e.  (  ._|_  `  ( L `  g )
)  /\  X  =/=  .0.  ) )
5853, 56, 57sylanbrc 645 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( (  ._|_  `  ( L `
 g ) ) 
\  {  .0.  }
) )
5914, 29, 15, 30, 33, 10, 11, 52, 28, 58, 51dochsnkrlem2 31729 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  g )
)  e.  (LSAtoms `  U
) )
6014, 29, 15, 30, 31, 32, 8, 9, 33, 10, 11, 12, 34, 35, 36, 16, 48lcfrlem15 31816 . . . . . . . . . 10  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  ( J `  X )
) ) )
61 eldifsn 3825 . . . . . . . . . 10  |-  ( X  e.  ( (  ._|_  `  ( L `  ( J `  X )
) )  \  {  .0.  } )  <->  ( X  e.  (  ._|_  `  ( L `  ( J `  X ) ) )  /\  X  =/=  .0.  ) )
6260, 55, 61sylanbrc 645 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  ( J `  X ) ) )  \  {  .0.  } ) )
6314, 29, 15, 30, 33, 10, 11, 16, 49, 62, 51dochsnkrlem2 31729 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( L `
 ( J `  X ) ) )  e.  (LSAtoms `  U
) )
64633ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  ( J `  X ) ) )  e.  (LSAtoms `  U
) )
65603ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( 
._|_  `  ( L `  ( J `  X ) ) ) )
6633, 51, 18, 59, 64, 56, 53, 65lsat2el 29266 . . . . . 6  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  g )
)  =  (  ._|_  `  ( L `  ( J `  X )
) ) )
67 eqid 2358 . . . . . . 7  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
68 lcfrlem16.gs . . . . . . . . . 10  |-  ( ph  ->  G  C_  C )
69683ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  G  C_  C
)
70 simp2 956 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  G
)
7169, 70sseldd 3257 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  C
)
7214, 67, 29, 15, 10, 11, 35, 52, 28lcfl5 31755 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( g  e.  C  <->  ( L `  g )  e.  ran  ( ( DIsoH `  K
) `  W )
) )
7371, 72mpbid 201 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  g )  e.  ran  ( ( DIsoH `  K
) `  W )
)
7414, 29, 15, 30, 31, 32, 8, 9, 33, 10, 11, 12, 34, 35, 36, 16, 48lcfrlem13 31814 . . . . . . . . . 10  |-  ( ph  ->  ( J `  X
)  e.  ( C 
\  { Q }
) )
75 eldifi 3374 . . . . . . . . . 10  |-  ( ( J `  X )  e.  ( C  \  { Q } )  -> 
( J `  X
)  e.  C )
7674, 75syl 15 . . . . . . . . 9  |-  ( ph  ->  ( J `  X
)  e.  C )
7714, 67, 29, 15, 10, 11, 35, 16, 49lcfl5 31755 . . . . . . . . 9  |-  ( ph  ->  ( ( J `  X )  e.  C  <->  ( L `  ( J `
 X ) )  e.  ran  ( (
DIsoH `  K ) `  W ) ) )
7876, 77mpbid 201 . . . . . . . 8  |-  ( ph  ->  ( L `  ( J `  X )
)  e.  ran  (
( DIsoH `  K ) `  W ) )
79783ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  ( J `  X ) )  e.  ran  (
( DIsoH `  K ) `  W ) )
8014, 67, 29, 52, 73, 79doch11 31632 . . . . . 6  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( (  ._|_  `  ( L `  g
) )  =  ( 
._|_  `  ( L `  ( J `  X ) ) )  <->  ( L `  g )  =  ( L `  ( J `
 X ) ) ) )
8166, 80mpbid 201 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  g )  =  ( L `  ( J `
 X ) ) )
828, 9, 10, 11, 12, 13, 18, 28, 50, 81eqlkr4 29424 . . . 4  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  E. k  e.  R  ( J `  X )  =  ( k ( .s `  D ) g ) )
83243ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  U  e.  LMod )
8483adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  U  e.  LMod )
85193ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  G  e.  P
)
8685adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  G  e.  P )
87 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  k  e.  R )
88 simpl2 959 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  g  e.  G )
898, 9, 12, 13, 21, 84, 86, 87, 88ldualssvscl 29417 . . . . . 6  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  (
k ( .s `  D ) g )  e.  G )
90 eleq1 2418 . . . . . 6  |-  ( ( J `  X )  =  ( k ( .s `  D ) g )  ->  (
( J `  X
)  e.  G  <->  ( k
( .s `  D
) g )  e.  G ) )
9189, 90syl5ibrcom 213 . . . . 5  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  (
( J `  X
)  =  ( k ( .s `  D
) g )  -> 
( J `  X
)  e.  G ) )
9291rexlimdva 2743 . . . 4  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( E. k  e.  R  ( J `  X )  =  ( k ( .s `  D ) g )  ->  ( J `  X )  e.  G
) )
9382, 92mpd 14 . . 3  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( J `  X )  e.  G
)
9493rexlimdv3a 2745 . 2  |-  ( ph  ->  ( E. g  e.  G  X  e.  ( 
._|_  `  ( L `  g ) )  -> 
( J `  X
)  e.  G ) )
957, 94mpd 14 1  |-  ( ph  ->  ( J `  X
)  e.  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   {crab 2623    \ cdif 3225    C_ wss 3228   {csn 3716   U_ciun 3986    e. cmpt 4158   ran crn 4772   ` cfv 5337  (class class class)co 5945   iota_crio 6384   Basecbs 13245   +g cplusg 13305  Scalarcsca 13308   .scvsca 13309   0gc0g 13499   LModclmod 15726   LSubSpclss 15788   LVecclvec 15954  LSAtomsclsa 29233  LFnlclfn 29316  LKerclk 29344  LDualcld 29382   HLchlt 29609   LHypclh 30242   DVecHcdvh 31337   DIsoHcdih 31487   ocHcoch 31606
This theorem is referenced by:  lcfrlem27  31828  lcfrlem37  31838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-fal 1320  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-tpos 6321  df-undef 6385  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-0g 13503  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-subg 14717  df-cntz 14892  df-lsm 15046  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-dvr 15564  df-drng 15613  df-lmod 15728  df-lss 15789  df-lsp 15828  df-lvec 15955  df-lsatoms 29235  df-lshyp 29236  df-lfl 29317  df-lkr 29345  df-ldual 29383  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756  df-lplanes 29757  df-lvols 29758  df-lines 29759  df-psubsp 29761  df-pmap 29762  df-padd 30054  df-lhyp 30246  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417  df-tgrp 31001  df-tendo 31013  df-edring 31015  df-dveca 31261  df-disoa 31288  df-dvech 31338  df-dib 31398  df-dic 31432  df-dih 31488  df-doch 31607  df-djh 31654
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