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Theorem lcfrlem16 36355
Description: Lemma for lcfr 36382. (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.  }
) )
21eldifad 3488 . . . 4  |-  ( ph  ->  X  e.  E )
3 lcfrlem16.m . . . 4  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
42, 3syl6eleq 2565 . . 3  |-  ( ph  ->  X  e.  U_ g  e.  G  (  ._|_  `  ( L `  g
) ) )
5 eliun 4330 . . 3  |-  ( X  e.  U_ g  e.  G  (  ._|_  `  ( L `  g )
)  <->  E. g  e.  G  X  e.  (  ._|_  `  ( L `  g
) ) )
64, 5sylib 196 . 2  |-  ( ph  ->  E. g  e.  G  X  e.  (  ._|_  `  ( L `  g
) ) )
7 lcf1o.s . . . . 5  |-  S  =  (Scalar `  U )
8 lcf1o.r . . . . 5  |-  R  =  ( Base `  S
)
9 lcf1o.f . . . . 5  |-  F  =  (LFnl `  U )
10 lcf1o.l . . . . 5  |-  L  =  (LKer `  U )
11 lcf1o.d . . . . 5  |-  D  =  (LDual `  U )
12 eqid 2467 . . . . 5  |-  ( .s
`  D )  =  ( .s `  D
)
13 lcf1o.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
14 lcf1o.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
15 lcflo.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
1613, 14, 15dvhlvec 35906 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
17163ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  U  e.  LVec )
18 lcfrlem16.g . . . . . . . 8  |-  ( ph  ->  G  e.  P )
19 eqid 2467 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
20 lcfrlem16.p . . . . . . . . 9  |-  P  =  ( LSubSp `  D )
2119, 20lssel 17364 . . . . . . . 8  |-  ( ( G  e.  P  /\  g  e.  G )  ->  g  e.  ( Base `  D ) )
2218, 21sylan 471 . . . . . . 7  |-  ( (
ph  /\  g  e.  G )  ->  g  e.  ( Base `  D
) )
2313, 14, 15dvhlmod 35907 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
249, 11, 19, 23ldualvbase 33923 . . . . . . . 8  |-  ( ph  ->  ( Base `  D
)  =  F )
2524adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  e.  G )  ->  ( Base `  D )  =  F )
2622, 25eleqtrd 2557 . . . . . 6  |-  ( (
ph  /\  g  e.  G )  ->  g  e.  F )
27263adant3 1016 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  F
)
28 lcf1o.o . . . . . . 7  |-  ._|_  =  ( ( ocH `  K
) `  W )
29 lcf1o.v . . . . . . 7  |-  V  =  ( Base `  U
)
30 lcf1o.a . . . . . . 7  |-  .+  =  ( +g  `  U )
31 lcf1o.t . . . . . . 7  |-  .x.  =  ( .s `  U )
32 lcf1o.z . . . . . . 7  |-  .0.  =  ( 0g `  U )
33 lcf1o.q . . . . . . 7  |-  Q  =  ( 0g `  D
)
34 lcf1o.c . . . . . . 7  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
35 lcf1o.j . . . . . . 7  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
3615adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
3723adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  g  e.  G )  ->  U  e.  LMod )
3829, 9, 10, 37, 26lkrssv 33893 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  G )  ->  ( L `  g )  C_  V )
3913, 14, 29, 28dochssv 36152 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  g )  C_  V
)  ->  (  ._|_  `  ( L `  g
) )  C_  V
)
4036, 38, 39syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  G )  ->  (  ._|_  `  ( L `  g ) )  C_  V )
4140ralrimiva 2878 . . . . . . . . . . 11  |-  ( ph  ->  A. g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
42 iunss 4366 . . . . . . . . . . 11  |-  ( U_ g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V 
<-> 
A. g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
4341, 42sylibr 212 . . . . . . . . . 10  |-  ( ph  ->  U_ g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
443, 43syl5eqss 3548 . . . . . . . . 9  |-  ( ph  ->  E  C_  V )
4544ssdifd 3640 . . . . . . . 8  |-  ( ph  ->  ( E  \  {  .0.  } )  C_  ( V  \  {  .0.  }
) )
4645, 1sseldd 3505 . . . . . . 7  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
4713, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46lcfrlem10 36349 . . . . . 6  |-  ( ph  ->  ( J `  X
)  e.  F )
48473ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( J `  X )  e.  F
)
49 eqid 2467 . . . . . . 7  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
50153ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
51 simp3 998 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( 
._|_  `  ( L `  g ) ) )
52 eldifsni 4153 . . . . . . . . . . 11  |-  ( X  e.  ( E  \  {  .0.  } )  ->  X  =/=  .0.  )
531, 52syl 16 . . . . . . . . . 10  |-  ( ph  ->  X  =/=  .0.  )
54533ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  =/=  .0.  )
55 eldifsn 4152 . . . . . . . . 9  |-  ( X  e.  ( (  ._|_  `  ( L `  g
) )  \  {  .0.  } )  <->  ( X  e.  (  ._|_  `  ( L `  g )
)  /\  X  =/=  .0.  ) )
5651, 54, 55sylanbrc 664 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( (  ._|_  `  ( L `
 g ) ) 
\  {  .0.  }
) )
5713, 28, 14, 29, 32, 9, 10, 50, 27, 56, 49dochsnkrlem2 36267 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  g )
)  e.  (LSAtoms `  U
) )
5813, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46lcfrlem15 36354 . . . . . . . . . 10  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  ( J `  X )
) ) )
59 eldifsn 4152 . . . . . . . . . 10  |-  ( X  e.  ( (  ._|_  `  ( L `  ( J `  X )
) )  \  {  .0.  } )  <->  ( X  e.  (  ._|_  `  ( L `  ( J `  X ) ) )  /\  X  =/=  .0.  ) )
6058, 53, 59sylanbrc 664 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  ( J `  X ) ) )  \  {  .0.  } ) )
6113, 28, 14, 29, 32, 9, 10, 15, 47, 60, 49dochsnkrlem2 36267 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( L `
 ( J `  X ) ) )  e.  (LSAtoms `  U
) )
62613ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  ( J `  X ) ) )  e.  (LSAtoms `  U
) )
63583ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( 
._|_  `  ( L `  ( J `  X ) ) ) )
6432, 49, 17, 57, 62, 54, 51, 63lsat2el 33804 . . . . . 6  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  g )
)  =  (  ._|_  `  ( L `  ( J `  X )
) ) )
65 eqid 2467 . . . . . . 7  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
66 lcfrlem16.gs . . . . . . . . . 10  |-  ( ph  ->  G  C_  C )
67663ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  G  C_  C
)
68 simp2 997 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  G
)
6967, 68sseldd 3505 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  C
)
7013, 65, 28, 14, 9, 10, 34, 50, 27lcfl5 36293 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( g  e.  C  <->  ( L `  g )  e.  ran  ( ( DIsoH `  K
) `  W )
) )
7169, 70mpbid 210 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  g )  e.  ran  ( ( DIsoH `  K
) `  W )
)
7213, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46lcfrlem13 36352 . . . . . . . . . 10  |-  ( ph  ->  ( J `  X
)  e.  ( C 
\  { Q }
) )
7372eldifad 3488 . . . . . . . . 9  |-  ( ph  ->  ( J `  X
)  e.  C )
7413, 65, 28, 14, 9, 10, 34, 15, 47lcfl5 36293 . . . . . . . . 9  |-  ( ph  ->  ( ( J `  X )  e.  C  <->  ( L `  ( J `
 X ) )  e.  ran  ( (
DIsoH `  K ) `  W ) ) )
7573, 74mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( L `  ( J `  X )
)  e.  ran  (
( DIsoH `  K ) `  W ) )
76753ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  ( J `  X ) )  e.  ran  (
( DIsoH `  K ) `  W ) )
7713, 65, 28, 50, 71, 76doch11 36170 . . . . . 6  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( (  ._|_  `  ( L `  g
) )  =  ( 
._|_  `  ( L `  ( J `  X ) ) )  <->  ( L `  g )  =  ( L `  ( J `
 X ) ) ) )
7864, 77mpbid 210 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  g )  =  ( L `  ( J `
 X ) ) )
797, 8, 9, 10, 11, 12, 17, 27, 48, 78eqlkr4 33962 . . . 4  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  E. k  e.  R  ( J `  X )  =  ( k ( .s `  D ) g ) )
80233ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  U  e.  LMod )
8180adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  U  e.  LMod )
82183ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  G  e.  P
)
8382adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  G  e.  P )
84 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  k  e.  R )
85 simpl2 1000 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  g  e.  G )
867, 8, 11, 12, 20, 81, 83, 84, 85ldualssvscl 33955 . . . . . 6  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  (
k ( .s `  D ) g )  e.  G )
87 eleq1 2539 . . . . . 6  |-  ( ( J `  X )  =  ( k ( .s `  D ) g )  ->  (
( J `  X
)  e.  G  <->  ( k
( .s `  D
) g )  e.  G ) )
8886, 87syl5ibrcom 222 . . . . 5  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  (
( J `  X
)  =  ( k ( .s `  D
) g )  -> 
( J `  X
)  e.  G ) )
8988rexlimdva 2955 . . . 4  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( E. k  e.  R  ( J `  X )  =  ( k ( .s `  D ) g )  ->  ( J `  X )  e.  G
) )
9079, 89mpd 15 . . 3  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( J `  X )  e.  G
)
9190rexlimdv3a 2957 . 2  |-  ( ph  ->  ( E. g  e.  G  X  e.  ( 
._|_  `  ( L `  g ) )  -> 
( J `  X
)  e.  G ) )
926, 91mpd 15 1  |-  ( ph  ->  ( J `  X
)  e.  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    \ cdif 3473    C_ wss 3476   {csn 4027   U_ciun 4325    |-> cmpt 4505   ran crn 5000   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14483   +g cplusg 14548  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   LModclmod 17292   LSubSpclss 17358   LVecclvec 17528  LSAtomsclsa 33771  LFnlclfn 33854  LKerclk 33882  LDualcld 33920   HLchlt 34147   LHypclh 34780   DVecHcdvh 35875   DIsoHcdih 36025   ocHcoch 36144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-0g 14690  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-cntz 16147  df-lsm 16449  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-drng 17178  df-lmod 17294  df-lss 17359  df-lsp 17398  df-lvec 17529  df-lsatoms 33773  df-lshyp 33774  df-lfl 33855  df-lkr 33883  df-ldual 33921  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tgrp 35539  df-tendo 35551  df-edring 35553  df-dveca 35799  df-disoa 35826  df-dvech 35876  df-dib 35936  df-dic 35970  df-dih 36026  df-doch 36145  df-djh 36192
This theorem is referenced by:  lcfrlem27  36366  lcfrlem37  36376
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