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Theorem lcfrlem16 35197
Description: Lemma for lcfr 35224. (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 3402 . . . 4  |-  ( ph  ->  X  e.  E )
3 lcfrlem16.m . . . 4  |-  E  = 
U_ g  e.  G  (  ._|_  `  ( L `  g ) )
42, 3syl6eleq 2559 . . 3  |-  ( ph  ->  X  e.  U_ g  e.  G  (  ._|_  `  ( L `  g
) ) )
5 eliun 4274 . . 3  |-  ( X  e.  U_ g  e.  G  (  ._|_  `  ( L `  g )
)  <->  E. g  e.  G  X  e.  (  ._|_  `  ( L `  g
) ) )
64, 5sylib 201 . 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 2471 . . . . 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 34748 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
17163ad2ant1 1051 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  U  e.  LVec )
18 lcfrlem16.g . . . . . . . 8  |-  ( ph  ->  G  e.  P )
19 eqid 2471 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
20 lcfrlem16.p . . . . . . . . 9  |-  P  =  ( LSubSp `  D )
2119, 20lssel 18239 . . . . . . . 8  |-  ( ( G  e.  P  /\  g  e.  G )  ->  g  e.  ( Base `  D ) )
2218, 21sylan 479 . . . . . . 7  |-  ( (
ph  /\  g  e.  G )  ->  g  e.  ( Base `  D
) )
2313, 14, 15dvhlmod 34749 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
249, 11, 19, 23ldualvbase 32763 . . . . . . . 8  |-  ( ph  ->  ( Base `  D
)  =  F )
2524adantr 472 . . . . . . 7  |-  ( (
ph  /\  g  e.  G )  ->  ( Base `  D )  =  F )
2622, 25eleqtrd 2551 . . . . . 6  |-  ( (
ph  /\  g  e.  G )  ->  g  e.  F )
27263adant3 1050 . . . . 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 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
3723adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  g  e.  G )  ->  U  e.  LMod )
3829, 9, 10, 37, 26lkrssv 32733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  g  e.  G )  ->  ( L `  g )  C_  V )
3913, 14, 29, 28dochssv 34994 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  g )  C_  V
)  ->  (  ._|_  `  ( L `  g
) )  C_  V
)
4036, 38, 39syl2anc 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  G )  ->  (  ._|_  `  ( L `  g ) )  C_  V )
4140ralrimiva 2809 . . . . . . . . . . 11  |-  ( ph  ->  A. g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
42 iunss 4310 . . . . . . . . . . 11  |-  ( U_ g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V 
<-> 
A. g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
4341, 42sylibr 217 . . . . . . . . . 10  |-  ( ph  ->  U_ g  e.  G  (  ._|_  `  ( L `  g ) )  C_  V )
443, 43syl5eqss 3462 . . . . . . . . 9  |-  ( ph  ->  E  C_  V )
4544ssdifd 3558 . . . . . . . 8  |-  ( ph  ->  ( E  \  {  .0.  } )  C_  ( V  \  {  .0.  }
) )
4645, 1sseldd 3419 . . . . . . 7  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
4713, 28, 14, 29, 30, 31, 7, 8, 32, 9, 10, 11, 33, 34, 35, 15, 46lcfrlem10 35191 . . . . . 6  |-  ( ph  ->  ( J `  X
)  e.  F )
48473ad2ant1 1051 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( J `  X )  e.  F
)
49 eqid 2471 . . . . . . 7  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
50153ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
51 simp3 1032 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( 
._|_  `  ( L `  g ) ) )
52 eldifsni 4089 . . . . . . . . . . 11  |-  ( X  e.  ( E  \  {  .0.  } )  ->  X  =/=  .0.  )
531, 52syl 17 . . . . . . . . . 10  |-  ( ph  ->  X  =/=  .0.  )
54533ad2ant1 1051 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  =/=  .0.  )
55 eldifsn 4088 . . . . . . . . 9  |-  ( X  e.  ( (  ._|_  `  ( L `  g
) )  \  {  .0.  } )  <->  ( X  e.  (  ._|_  `  ( L `  g )
)  /\  X  =/=  .0.  ) )
5651, 54, 55sylanbrc 677 . . . . . . . 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 35109 . . . . . . 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 35196 . . . . . . . . . 10  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  ( J `  X )
) ) )
59 eldifsn 4088 . . . . . . . . . 10  |-  ( X  e.  ( (  ._|_  `  ( L `  ( J `  X )
) )  \  {  .0.  } )  <->  ( X  e.  (  ._|_  `  ( L `  ( J `  X ) ) )  /\  X  =/=  .0.  ) )
6058, 53, 59sylanbrc 677 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  ( J `  X ) ) )  \  {  .0.  } ) )
6113, 28, 14, 29, 32, 9, 10, 15, 47, 60, 49dochsnkrlem2 35109 . . . . . . . 8  |-  ( ph  ->  (  ._|_  `  ( L `
 ( J `  X ) ) )  e.  (LSAtoms `  U
) )
62613ad2ant1 1051 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  ( J `  X ) ) )  e.  (LSAtoms `  U
) )
63583ad2ant1 1051 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  X  e.  ( 
._|_  `  ( L `  ( J `  X ) ) ) )
6432, 49, 17, 57, 62, 54, 51, 63lsat2el 32644 . . . . . 6  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  (  ._|_  `  ( L `  g )
)  =  (  ._|_  `  ( L `  ( J `  X )
) ) )
65 eqid 2471 . . . . . . 7  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
66 lcfrlem16.gs . . . . . . . . . 10  |-  ( ph  ->  G  C_  C )
67663ad2ant1 1051 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  G  C_  C
)
68 simp2 1031 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  G
)
6967, 68sseldd 3419 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  g  e.  C
)
7013, 65, 28, 14, 9, 10, 34, 50, 27lcfl5 35135 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( g  e.  C  <->  ( L `  g )  e.  ran  ( ( DIsoH `  K
) `  W )
) )
7169, 70mpbid 215 . . . . . . 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 35194 . . . . . . . . . 10  |-  ( ph  ->  ( J `  X
)  e.  ( C 
\  { Q }
) )
7372eldifad 3402 . . . . . . . . 9  |-  ( ph  ->  ( J `  X
)  e.  C )
7413, 65, 28, 14, 9, 10, 34, 15, 47lcfl5 35135 . . . . . . . . 9  |-  ( ph  ->  ( ( J `  X )  e.  C  <->  ( L `  ( J `
 X ) )  e.  ran  ( (
DIsoH `  K ) `  W ) ) )
7573, 74mpbid 215 . . . . . . . 8  |-  ( ph  ->  ( L `  ( J `  X )
)  e.  ran  (
( DIsoH `  K ) `  W ) )
76753ad2ant1 1051 . . . . . . 7  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  ( J `  X ) )  e.  ran  (
( DIsoH `  K ) `  W ) )
7713, 65, 28, 50, 71, 76doch11 35012 . . . . . 6  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( (  ._|_  `  ( L `  g
) )  =  ( 
._|_  `  ( L `  ( J `  X ) ) )  <->  ( L `  g )  =  ( L `  ( J `
 X ) ) ) )
7864, 77mpbid 215 . . . . 5  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  ( L `  g )  =  ( L `  ( J `
 X ) ) )
797, 8, 9, 10, 11, 12, 17, 27, 48, 78eqlkr4 32802 . . . 4  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  E. k  e.  R  ( J `  X )  =  ( k ( .s `  D ) g ) )
80233ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  U  e.  LMod )
8180adantr 472 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  U  e.  LMod )
82183ad2ant1 1051 . . . . . . . 8  |-  ( (
ph  /\  g  e.  G  /\  X  e.  ( 
._|_  `  ( L `  g ) ) )  ->  G  e.  P
)
8382adantr 472 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  G  e.  P )
84 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  k  e.  R )
85 simpl2 1034 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  g  e.  G )
867, 8, 11, 12, 20, 81, 83, 84, 85ldualssvscl 32795 . . . . . 6  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  (
k ( .s `  D ) g )  e.  G )
87 eleq1 2537 . . . . . 6  |-  ( ( J `  X )  =  ( k ( .s `  D ) g )  ->  (
( J `  X
)  e.  G  <->  ( k
( .s `  D
) g )  e.  G ) )
8886, 87syl5ibrcom 230 . . . . 5  |-  ( ( ( ph  /\  g  e.  G  /\  X  e.  (  ._|_  `  ( L `
 g ) ) )  /\  k  e.  R )  ->  (
( J `  X
)  =  ( k ( .s `  D
) g )  -> 
( J `  X
)  e.  G ) )
8988rexlimdva 2871 . . . 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 2873 . 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760    \ cdif 3387    C_ wss 3390   {csn 3959   U_ciun 4269    |-> cmpt 4454   ran crn 4840   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   +g cplusg 15268  Scalarcsca 15271   .scvsca 15272   0gc0g 15416   LModclmod 18169   LSubSpclss 18233   LVecclvec 18403  LSAtomsclsa 32611  LFnlclfn 32694  LKerclk 32722  LDualcld 32760   HLchlt 32987   LHypclh 33620   DVecHcdvh 34717   DIsoHcdih 34867   ocHcoch 34986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-undef 7038  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-0g 15418  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-cntz 17049  df-lsm 17366  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-lmod 18171  df-lss 18234  df-lsp 18273  df-lvec 18404  df-lsatoms 32613  df-lshyp 32614  df-lfl 32695  df-lkr 32723  df-ldual 32761  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tgrp 34381  df-tendo 34393  df-edring 34395  df-dveca 34641  df-disoa 34668  df-dvech 34718  df-dib 34778  df-dic 34812  df-dih 34868  df-doch 34987  df-djh 35034
This theorem is referenced by:  lcfrlem27  35208  lcfrlem37  35218
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