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Theorem lsmsat 29868
Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 30664 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
Hypotheses
Ref Expression
lsmsat.o  |-  .0.  =  ( 0g `  W )
lsmsat.s  |-  S  =  ( LSubSp `  W )
lsmsat.p  |-  .(+)  =  (
LSSum `  W )
lsmsat.a  |-  A  =  (LSAtoms `  W )
lsmsat.w  |-  ( ph  ->  W  e.  LMod )
lsmsat.t  |-  ( ph  ->  T  e.  S )
lsmsat.u  |-  ( ph  ->  U  e.  S )
lsmsat.q  |-  ( ph  ->  Q  e.  A )
lsmsat.n  |-  ( ph  ->  T  =/=  {  .0.  } )
lsmsat.l  |-  ( ph  ->  Q  C_  ( T  .(+) 
U ) )
Assertion
Ref Expression
lsmsat  |-  ( ph  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) )
Distinct variable groups:    A, p    .(+) ,
p    Q, p    T, p    U, p    W, p
Allowed substitution hints:    ph( p)    S( p)    .0. ( p)

Proof of Theorem lsmsat
Dummy variables  q 
r  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmsat.q . . 3  |-  ( ph  ->  Q  e.  A )
2 lsmsat.w . . . 4  |-  ( ph  ->  W  e.  LMod )
3 eqid 2438 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2438 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsat.o . . . . 5  |-  .0.  =  ( 0g `  W )
6 lsmsat.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 29851 . . . 4  |-  ( W  e.  LMod  ->  ( Q  e.  A  <->  E. r  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { r } ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  ( Q  e.  A  <->  E. r  e.  ( (
Base `  W )  \  {  .0.  } ) Q  =  ( (
LSpan `  W ) `  { r } ) ) )
91, 8mpbid 203 . 2  |-  ( ph  ->  E. r  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { r } ) )
10 simp3 960 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  Q  =  ( ( LSpan `  W ) `  { r } ) )
11 lsmsat.l . . . . . . . . . 10  |-  ( ph  ->  Q  C_  ( T  .(+) 
U ) )
12113ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  Q  C_  ( T  .(+)  U ) )
1310, 12eqsstr3d 3385 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( ( LSpan `  W
) `  { r } )  C_  ( T  .(+)  U ) )
14 lsmsat.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
1523ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  W  e.  LMod )
16 lsmsat.t . . . . . . . . . . 11  |-  ( ph  ->  T  e.  S )
17 lsmsat.u . . . . . . . . . . 11  |-  ( ph  ->  U  e.  S )
18 lsmsat.p . . . . . . . . . . . 12  |-  .(+)  =  (
LSSum `  W )
1914, 18lsmcl 16157 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
202, 16, 17, 19syl3anc 1185 . . . . . . . . . 10  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
21203ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( T  .(+)  U )  e.  S )
22 eldifi 3471 . . . . . . . . . 10  |-  ( r  e.  ( ( Base `  W )  \  {  .0.  } )  ->  r  e.  ( Base `  W
) )
23223ad2ant2 980 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
r  e.  ( Base `  W ) )
243, 14, 4, 15, 21, 23lspsnel5 16073 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( r  e.  ( T  .(+)  U )  <->  ( ( LSpan `  W ) `  { r } ) 
C_  ( T  .(+)  U ) ) )
2513, 24mpbird 225 . . . . . . 7  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
r  e.  ( T 
.(+)  U ) )
2614lsssssubg 16036 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
2715, 26syl 16 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  S  C_  (SubGrp `  W
) )
28163ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  T  e.  S )
2927, 28sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  T  e.  (SubGrp `  W
) )
30173ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  U  e.  S )
3127, 30sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  U  e.  (SubGrp `  W
) )
32 eqid 2438 . . . . . . . . 9  |-  ( +g  `  W )  =  ( +g  `  W )
3332, 18lsmelval 15285 . . . . . . . 8  |-  ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  ( r  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  r  =  ( y
( +g  `  W ) z ) ) )
3429, 31, 33syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( r  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  r  =  ( y ( +g  `  W ) z ) ) )
3525, 34mpbid 203 . . . . . 6  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  E. y  e.  T  E. z  e.  U  r  =  ( y
( +g  `  W ) z ) )
36 lsmsat.n . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  =/=  {  .0.  } )
375, 14lssne0 16029 . . . . . . . . . . . . . . . 16  |-  ( T  e.  S  ->  ( T  =/=  {  .0.  }  <->  E. q  e.  T  q  =/=  .0.  ) )
3816, 37syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( T  =/=  {  .0.  }  <->  E. q  e.  T  q  =/=  .0.  ) )
3936, 38mpbid 203 . . . . . . . . . . . . . 14  |-  ( ph  ->  E. q  e.  T  q  =/=  .0.  )
4039adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  E. q  e.  T  q  =/=  .0.  )
41403ad2ant1 979 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  E. q  e.  T  q  =/=  .0.  )
4241adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =  .0.  )  ->  E. q  e.  T  q  =/=  .0.  )
432adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LMod )
44433ad2ant1 979 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  W  e.  LMod )
4544adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  W  e.  LMod )
4616adantr 453 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  T  e.  S )
47463ad2ant1 979 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  T  e.  S
)
4847adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  T  e.  S )
49 simpr2 965 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  q  e.  T )
503, 14lssel 16016 . . . . . . . . . . . . . . . . 17  |-  ( ( T  e.  S  /\  q  e.  T )  ->  q  e.  ( Base `  W ) )
5148, 49, 50syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  q  e.  ( Base `  W
) )
52 simpr3 966 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  q  =/=  .0.  )
533, 4, 5, 6lsatlspsn2 29852 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  q  e.  ( Base `  W
)  /\  q  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { q } )  e.  A )
5445, 51, 52, 53syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { q } )  e.  A )
5514, 4, 45, 48, 49lspsnel5a 16074 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { q } ) 
C_  T )
56 simpl3 963 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  r  =  ( y ( +g  `  W ) z ) )
57 simpr1 964 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  y  =  .0.  )
5857oveq1d 6098 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
y ( +g  `  W
) z )  =  (  .0.  ( +g  `  W ) z ) )
5917adantr 453 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  U  e.  S )
60593ad2ant1 979 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  U  e.  S
)
61 simp2r 985 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  z  e.  U
)
623, 14lssel 16016 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( U  e.  S  /\  z  e.  U )  ->  z  e.  ( Base `  W ) )
6360, 61, 62syl2anc 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  z  e.  (
Base `  W )
)
6463adantr 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  z  e.  ( Base `  W
) )
653, 32, 5lmod0vlid 15982 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( W  e.  LMod  /\  z  e.  ( Base `  W
) )  ->  (  .0.  ( +g  `  W
) z )  =  z )
6645, 64, 65syl2anc 644 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (  .0.  ( +g  `  W
) z )  =  z )
6756, 58, 663eqtrd 2474 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  r  =  z )
6867sneqd 3829 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  { r }  =  { z } )
6968fveq2d 5734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { r } )  =  ( ( LSpan `  W ) `  {
z } ) )
7014, 4, 44, 60, 61lspsnel5a 16074 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
z } )  C_  U )
7170adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { z } ) 
C_  U )
7269, 71eqsstrd 3384 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { r } ) 
C_  U )
733, 4lspsnsubg 16058 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e.  LMod  /\  q  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { q } )  e.  (SubGrp `  W
) )
7445, 51, 73syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { q } )  e.  (SubGrp `  W
) )
7545, 26syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  S  C_  (SubGrp `  W )
)
7660adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  U  e.  S )
7775, 76sseldd 3351 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  U  e.  (SubGrp `  W )
)
7818lsmub2 15293 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( LSpan `  W
) `  { q } )  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  (
( ( LSpan `  W
) `  { q } )  .(+)  U ) )
7974, 77, 78syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  U  C_  ( ( ( LSpan `  W ) `  {
q } )  .(+)  U ) )
8072, 79sstrd 3360 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  (
( LSpan `  W ) `  { r } ) 
C_  ( ( (
LSpan `  W ) `  { q } ) 
.(+)  U ) )
81 sseq1 3371 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( p  C_  T  <->  ( ( LSpan `  W ) `  { q } ) 
C_  T ) )
82 oveq1 6090 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( p  .(+)  U )  =  ( ( (
LSpan `  W ) `  { q } ) 
.(+)  U ) )
8382sseq2d 3378 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U )  <-> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { q } )  .(+)  U ) ) )
8481, 83anbi12d 693 . . . . . . . . . . . . . . . 16  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( ( p  C_  T  /\  ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U ) )  <->  ( ( (
LSpan `  W ) `  { q } ) 
C_  T  /\  (
( LSpan `  W ) `  { r } ) 
C_  ( ( (
LSpan `  W ) `  { q } ) 
.(+)  U ) ) ) )
8584rspcev 3054 . . . . . . . . . . . . . . 15  |-  ( ( ( ( LSpan `  W
) `  { q } )  e.  A  /\  ( ( ( LSpan `  W ) `  {
q } )  C_  T  /\  ( ( LSpan `  W ) `  {
r } )  C_  ( ( ( LSpan `  W ) `  {
q } )  .(+)  U ) ) )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) )
8654, 55, 80, 85syl12anc 1183 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  ( y  =  .0. 
/\  q  e.  T  /\  q  =/=  .0.  ) )  ->  E. p  e.  A  ( p  C_  T  /\  ( (
LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) )
87863exp2 1172 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( y  =  .0.  ->  ( q  e.  T  ->  ( q  =/=  .0.  ->  E. p  e.  A  ( p  C_  T  /\  ( (
LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) ) ) ) )
8887imp 420 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =  .0.  )  ->  ( q  e.  T  ->  ( q  =/=  .0.  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) ) )
8988rexlimdv 2831 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =  .0.  )  ->  ( E. q  e.  T  q  =/=  .0.  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
9042, 89mpd 15 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =  .0.  )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) )
9144adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  ->  W  e.  LMod )
92 simp2l 984 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  y  e.  T
)
933, 14lssel 16016 . . . . . . . . . . . . . 14  |-  ( ( T  e.  S  /\  y  e.  T )  ->  y  e.  ( Base `  W ) )
9447, 92, 93syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  y  e.  (
Base `  W )
)
9594adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  -> 
y  e.  ( Base `  W ) )
96 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  -> 
y  =/=  .0.  )
973, 4, 5, 6lsatlspsn2 29852 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
)  /\  y  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { y } )  e.  A )
9891, 95, 96, 97syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  -> 
( ( LSpan `  W
) `  { y } )  e.  A
)
9914, 4, 44, 47, 92lspsnel5a 16074 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
y } )  C_  T )
10099adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  -> 
( ( LSpan `  W
) `  { y } )  C_  T
)
101 simp3 960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  r  =  ( y ( +g  `  W
) z ) )
102101sneqd 3829 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  { r }  =  { ( y ( +g  `  W
) z ) } )
103102fveq2d 5734 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
r } )  =  ( ( LSpan `  W
) `  { (
y ( +g  `  W
) z ) } ) )
1043, 32, 4lspvadd 16170 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
)  /\  z  e.  ( Base `  W )
)  ->  ( ( LSpan `  W ) `  { ( y ( +g  `  W ) z ) } ) 
C_  ( ( LSpan `  W ) `  {
y ,  z } ) )
10544, 94, 63, 104syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
( y ( +g  `  W ) z ) } )  C_  (
( LSpan `  W ) `  { y ,  z } ) )
106103, 105eqsstrd 3384 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
r } )  C_  ( ( LSpan `  W
) `  { y ,  z } ) )
1073, 4, 18, 44, 94, 63lsmpr 16163 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
y ,  z } )  =  ( ( ( LSpan `  W ) `  { y } ) 
.(+)  ( ( LSpan `  W ) `  {
z } ) ) )
108106, 107sseqtrd 3386 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
r } )  C_  ( ( ( LSpan `  W ) `  {
y } )  .(+)  ( ( LSpan `  W ) `  { z } ) ) )
10944, 26syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  S  C_  (SubGrp `  W ) )
1103, 14, 4lspsncl 16055 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { y } )  e.  S )
11144, 94, 110syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
y } )  e.  S )
112109, 111sseldd 3351 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
y } )  e.  (SubGrp `  W )
)
113109, 60sseldd 3351 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  U  e.  (SubGrp `  W ) )
11418lsmless2 15296 . . . . . . . . . . . . . 14  |-  ( ( ( ( LSpan `  W
) `  { y } )  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )  /\  ( ( LSpan `  W
) `  { z } )  C_  U
)  ->  ( (
( LSpan `  W ) `  { y } ) 
.(+)  ( ( LSpan `  W ) `  {
z } ) ) 
C_  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  U ) )
115112, 113, 70, 114syl3anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  ( ( LSpan `  W ) `  {
z } ) ) 
C_  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  U ) )
116108, 115sstrd 3360 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
r } )  C_  ( ( ( LSpan `  W ) `  {
y } )  .(+)  U ) )
117116adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  -> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { y } )  .(+)  U ) )
118 sseq1 3371 . . . . . . . . . . . . 13  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( p  C_  T  <->  ( ( LSpan `  W ) `  { y } ) 
C_  T ) )
119 oveq1 6090 . . . . . . . . . . . . . 14  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( p  .(+)  U )  =  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  U ) )
120119sseq2d 3378 . . . . . . . . . . . . 13  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U )  <-> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { y } )  .(+)  U ) ) )
121118, 120anbi12d 693 . . . . . . . . . . . 12  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( ( p  C_  T  /\  ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U ) )  <->  ( ( (
LSpan `  W ) `  { y } ) 
C_  T  /\  (
( LSpan `  W ) `  { r } ) 
C_  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  U ) ) ) )
122121rspcev 3054 . . . . . . . . . . 11  |-  ( ( ( ( LSpan `  W
) `  { y } )  e.  A  /\  ( ( ( LSpan `  W ) `  {
y } )  C_  T  /\  ( ( LSpan `  W ) `  {
r } )  C_  ( ( ( LSpan `  W ) `  {
y } )  .(+)  U ) ) )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) )
12398, 100, 117, 122syl12anc 1183 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( ( Base `  W )  \  {  .0.  } ) )  /\  ( y  e.  T  /\  z  e.  U )  /\  r  =  ( y ( +g  `  W ) z ) )  /\  y  =/=  .0.  )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) )
12490, 123pm2.61dane 2684 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) )
1251243exp 1153 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  (
( y  e.  T  /\  z  e.  U
)  ->  ( r  =  ( y ( +g  `  W ) z )  ->  E. p  e.  A  ( p  C_  T  /\  ( (
LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) ) ) )
126125rexlimdvv 2838 . . . . . . 7  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  ( E. y  e.  T  E. z  e.  U  r  =  ( y
( +g  `  W ) z )  ->  E. p  e.  A  ( p  C_  T  /\  ( (
LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) ) )
1271263adant3 978 . . . . . 6  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( E. y  e.  T  E. z  e.  U  r  =  ( y ( +g  `  W
) z )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
12835, 127mpd 15 . . . . 5  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) )
129 sseq1 3371 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( Q  C_  (
p  .(+)  U )  <->  ( ( LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) )
130129anbi2d 686 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( ( p  C_  T  /\  Q  C_  (
p  .(+)  U ) )  <-> 
( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
131130rexbidv 2728 . . . . . 6  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( E. p  e.  A  ( p  C_  T  /\  Q  C_  (
p  .(+)  U ) )  <->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
1321313ad2ant3 981 . . . . 5  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( E. p  e.  A  ( p  C_  T  /\  Q  C_  (
p  .(+)  U ) )  <->  E. p  e.  A  ( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
133128, 132mpbird 225 . . . 4  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) )
1341333exp 1153 . . 3  |-  ( ph  ->  ( r  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( Q  =  ( ( LSpan `  W ) `  {
r } )  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) ) ) )
135134rexlimdv 2831 . 2  |-  ( ph  ->  ( E. r  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { r } )  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) ) )
1369, 135mpd 15 1  |-  ( ph  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319    C_ wss 3322   {csn 3816   {cpr 3817   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725  SubGrpcsubg 14940   LSSumclsm 15270   LModclmod 15952   LSubSpclss 16010   LSpanclspn 16049  LSAtomsclsa 29834
This theorem is referenced by:  dochexmidlem4  32323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lsatoms 29836
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