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Theorem lsmsat 32026
Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 32822 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 2402 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2402 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsat.o . . . . 5  |-  .0.  =  ( 0g `  W )
6 lsmsat.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 32009 . . . 4  |-  ( W  e.  LMod  ->  ( Q  e.  A  <->  E. r  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { r } ) ) )
82, 7syl 17 . . 3  |-  ( ph  ->  ( Q  e.  A  <->  E. r  e.  ( (
Base `  W )  \  {  .0.  } ) Q  =  ( (
LSpan `  W ) `  { r } ) ) )
91, 8mpbid 210 . 2  |-  ( ph  ->  E. r  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { r } ) )
10 simp3 999 . . . . . . . . 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 1018 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  Q  C_  ( T  .(+)  U ) )
1310, 12eqsstr3d 3477 . . . . . . . 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 1018 . . . . . . . . 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 18049 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
202, 16, 17, 19syl3anc 1230 . . . . . . . . . 10  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
21203ad2ant1 1018 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( T  .(+)  U )  e.  S )
22 eldifi 3565 . . . . . . . . . 10  |-  ( r  e.  ( ( Base `  W )  \  {  .0.  } )  ->  r  e.  ( Base `  W
) )
23223ad2ant2 1019 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
r  e.  ( Base `  W ) )
243, 14, 4, 15, 21, 23lspsnel5 17961 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( r  e.  ( T  .(+)  U )  <->  ( ( LSpan `  W ) `  { r } ) 
C_  ( T  .(+)  U ) ) )
2513, 24mpbird 232 . . . . . . 7  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
r  e.  ( T 
.(+)  U ) )
2614lsssssubg 17924 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
2715, 26syl 17 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  S  C_  (SubGrp `  W
) )
28163ad2ant1 1018 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  T  e.  S )
2927, 28sseldd 3443 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  T  e.  (SubGrp `  W
) )
30173ad2ant1 1018 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  U  e.  S )
3127, 30sseldd 3443 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  U  e.  (SubGrp `  W
) )
32 eqid 2402 . . . . . . . . 9  |-  ( +g  `  W )  =  ( +g  `  W )
3332, 18lsmelval 16993 . . . . . . . 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 659 . . . . . . 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 210 . . . . . 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 17917 . . . . . . . . . . . . . . . 16  |-  ( T  e.  S  ->  ( T  =/=  {  .0.  }  <->  E. q  e.  T  q  =/=  .0.  ) )
3816, 37syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( T  =/=  {  .0.  }  <->  E. q  e.  T  q  =/=  .0.  ) )
3936, 38mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  E. q  e.  T  q  =/=  .0.  )
4039adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  E. q  e.  T  q  =/=  .0.  )
41403ad2ant1 1018 . . . . . . . . . . . 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 463 . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LMod )
44433ad2ant1 1018 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  W  e.  LMod )
4544adantr 463 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  T  e.  S )
47463ad2ant1 1018 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  T  e.  S
)
4847adantr 463 . . . . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . . . . 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 17904 . . . . . . . . . . . . . . . . 17  |-  ( ( T  e.  S  /\  q  e.  T )  ->  q  e.  ( Base `  W ) )
5148, 49, 50syl2anc 659 . . . . . . . . . . . . . . . 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 1005 . . . . . . . . . . . . . . . 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 32010 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  q  e.  ( Base `  W
)  /\  q  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { q } )  e.  A )
5445, 51, 52, 53syl3anc 1230 . . . . . . . . . . . . . . 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 17962 . . . . . . . . . . . . . . 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 1002 . . . . . . . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  U  e.  S )
60593ad2ant1 1018 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  U  e.  S
)
61 simp2r 1024 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  z  e.  U
)
623, 14lssel 17904 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( U  e.  S  /\  z  e.  U )  ->  z  e.  ( Base `  W ) )
6360, 61, 62syl2anc 659 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  z  e.  (
Base `  W )
)
6463adantr 463 . . . . . . . . . . . . . . . . . . . . 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 17862 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( W  e.  LMod  /\  z  e.  ( Base `  W
) )  ->  (  .0.  ( +g  `  W
) z )  =  z )
6645, 64, 65syl2anc 659 . . . . . . . . . . . . . . . . . . . 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 2447 . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . 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 5853 . . . . . . . . . . . . . . . . 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 17962 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
z } )  C_  U )
7170adantr 463 . . . . . . . . . . . . . . . . 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 3476 . . . . . . . . . . . . . . . 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 17946 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e.  LMod  /\  q  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { q } )  e.  (SubGrp `  W
) )
7445, 51, 73syl2anc 659 . . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . 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 3443 . . . . . . . . . . . . . . . . 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 17001 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( LSpan `  W
) `  { q } )  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  (
( ( LSpan `  W
) `  { q } )  .(+)  U ) )
7974, 77, 78syl2anc 659 . . . . . . . . . . . . . . . 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 3452 . . . . . . . . . . . . . . 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 3463 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( p  C_  T  <->  ( ( LSpan `  W ) `  { q } ) 
C_  T ) )
82 oveq1 6285 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( p  .(+)  U )  =  ( ( (
LSpan `  W ) `  { q } ) 
.(+)  U ) )
8382sseq2d 3470 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U )  <-> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { q } )  .(+)  U ) ) )
8481, 83anbi12d 709 . . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . 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 427 . . . . . . . . . . . 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 2894 . . . . . . . . . . 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 463 . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  y  e.  T
)
933, 14lssel 17904 . . . . . . . . . . . . . 14  |-  ( ( T  e.  S  /\  y  e.  T )  ->  y  e.  ( Base `  W ) )
9447, 92, 93syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  y  e.  (
Base `  W )
)
9594adantr 463 . . . . . . . . . . . 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 459 . . . . . . . . . . . 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 32010 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
)  /\  y  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { y } )  e.  A )
9891, 95, 96, 97syl3anc 1230 . . . . . . . . . . 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 17962 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
y } )  C_  T )
10099adantr 463 . . . . . . . . . . 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 999 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  r  =  ( y ( +g  `  W
) z ) )
102101sneqd 3984 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  { r }  =  { ( y ( +g  `  W
) z ) } )
103102fveq2d 5853 . . . . . . . . . . . . . . 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 18062 . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . 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 3476 . . . . . . . . . . . . . 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 18055 . . . . . . . . . . . . . 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 3478 . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . 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 17943 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { y } )  e.  S )
11144, 94, 110syl2anc 659 . . . . . . . . . . . . . . 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 3443 . . . . . . . . . . . . . 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 3443 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  U  e.  (SubGrp `  W ) )
11418lsmless2 17004 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 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 3452 . . . . . . . . . . . 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 463 . . . . . . . . . . 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 3463 . . . . . . . . . . . . 13  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( p  C_  T  <->  ( ( LSpan `  W ) `  { y } ) 
C_  T ) )
119 oveq1 6285 . . . . . . . . . . . . . 14  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( p  .(+)  U )  =  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  U ) )
120119sseq2d 3470 . . . . . . . . . . . . 13  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U )  <-> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { y } )  .(+)  U ) ) )
121118, 120anbi12d 709 . . . . . . . . . . . 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 3160 . . . . . . . . . . 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 1228 . . . . . . . . . 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 2721 . . . . . . . . 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 1196 . . . . . . . 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 2902 . . . . . . 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 1017 . . . . . 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 3463 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( Q  C_  (
p  .(+)  U )  <->  ( ( LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) )
130129anbi2d 702 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( ( p  C_  T  /\  Q  C_  (
p  .(+)  U ) )  <-> 
( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
131130rexbidv 2918 . . . . . 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 1020 . . . . 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 232 . . . 4  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) )
1341333exp 1196 . . 3  |-  ( ph  ->  ( r  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( Q  =  ( ( LSpan `  W ) `  {
r } )  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) ) ) )
135134rexlimdv 2894 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    \ cdif 3411    C_ wss 3414   {csn 3972   {cpr 3974   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   0gc0g 15054  SubGrpcsubg 16519   LSSumclsm 16978   LModclmod 17832   LSubSpclss 17898   LSpanclspn 17937  LSAtomsclsa 31992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-cntz 16679  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-lmod 17834  df-lss 17899  df-lsp 17938  df-lsatoms 31994
This theorem is referenced by:  dochexmidlem4  34483
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