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Theorem lsmsat 32375
Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 33171 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 2441 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2441 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lsmsat.o . . . . 5  |-  .0.  =  ( 0g `  W )
6 lsmsat.a . . . . 5  |-  A  =  (LSAtoms `  W )
73, 4, 5, 6islsat 32358 . . . 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 210 . 2  |-  ( ph  ->  E. r  e.  ( ( Base `  W
)  \  {  .0.  } ) Q  =  ( ( LSpan `  W ) `  { r } ) )
10 simp3 985 . . . . . . . . 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 1004 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  Q  C_  ( T  .(+)  U ) )
1310, 12eqsstr3d 3388 . . . . . . . 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 1004 . . . . . . . . 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 17142 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
202, 16, 17, 19syl3anc 1213 . . . . . . . . . 10  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
21203ad2ant1 1004 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
( T  .(+)  U )  e.  S )
22 eldifi 3475 . . . . . . . . . 10  |-  ( r  e.  ( ( Base `  W )  \  {  .0.  } )  ->  r  e.  ( Base `  W
) )
23223ad2ant2 1005 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  -> 
r  e.  ( Base `  W ) )
243, 14, 4, 15, 21, 23lspsnel5 17054 . . . . . . . 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 17017 . . . . . . . . . 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 1004 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  T  e.  S )
2927, 28sseldd 3354 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  T  e.  (SubGrp `  W
) )
30173ad2ant1 1004 . . . . . . . . 9  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  U  e.  S )
3127, 30sseldd 3354 . . . . . . . 8  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  Q  =  ( ( LSpan `  W
) `  { r } ) )  ->  U  e.  (SubGrp `  W
) )
32 eqid 2441 . . . . . . . . 9  |-  ( +g  `  W )  =  ( +g  `  W )
3332, 18lsmelval 16141 . . . . . . . 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 656 . . . . . . 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 17010 . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  E. q  e.  T  q  =/=  .0.  )
4039adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  E. q  e.  T  q  =/=  .0.  )
41403ad2ant1 1004 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LMod )
44433ad2ant1 1004 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  W  e.  LMod )
4544adantr 462 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  T  e.  S )
47463ad2ant1 1004 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  T  e.  S
)
4847adantr 462 . . . . . . . . . . . . . . . . 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 990 . . . . . . . . . . . . . . . . 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 16997 . . . . . . . . . . . . . . . . 17  |-  ( ( T  e.  S  /\  q  e.  T )  ->  q  e.  ( Base `  W ) )
5148, 49, 50syl2anc 656 . . . . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . . . 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 32359 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  q  e.  ( Base `  W
)  /\  q  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { q } )  e.  A )
5445, 51, 52, 53syl3anc 1213 . . . . . . . . . . . . . . 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 17055 . . . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  U  e.  S )
60593ad2ant1 1004 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  U  e.  S
)
61 simp2r 1010 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  z  e.  U
)
623, 14lssel 16997 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( U  e.  S  /\  z  e.  U )  ->  z  e.  ( Base `  W ) )
6360, 61, 62syl2anc 656 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  z  e.  (
Base `  W )
)
6463adantr 462 . . . . . . . . . . . . . . . . . . . . 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 16958 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( W  e.  LMod  /\  z  e.  ( Base `  W
) )  ->  (  .0.  ( +g  `  W
) z )  =  z )
6645, 64, 65syl2anc 656 . . . . . . . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . . . . . . 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 3886 . . . . . . . . . . . . . . . . . 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 5692 . . . . . . . . . . . . . . . . 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 17055 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
z } )  C_  U )
7170adantr 462 . . . . . . . . . . . . . . . . 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 3387 . . . . . . . . . . . . . . . 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 17039 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e.  LMod  /\  q  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { q } )  e.  (SubGrp `  W
) )
7445, 51, 73syl2anc 656 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . . . 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 16149 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( LSpan `  W
) `  { q } )  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  (
( ( LSpan `  W
) `  { q } )  .(+)  U ) )
7974, 77, 78syl2anc 656 . . . . . . . . . . . . . . . 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 3363 . . . . . . . . . . . . . . 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 3374 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( p  C_  T  <->  ( ( LSpan `  W ) `  { q } ) 
C_  T ) )
82 oveq1 6097 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( p  .(+)  U )  =  ( ( (
LSpan `  W ) `  { q } ) 
.(+)  U ) )
8382sseq2d 3381 . . . . . . . . . . . . . . . . 17  |-  ( p  =  ( ( LSpan `  W ) `  {
q } )  -> 
( ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U )  <-> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { q } )  .(+)  U ) ) )
8481, 83anbi12d 705 . . . . . . . . . . . . . . . 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 3070 . . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . 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 429 . . . . . . . . . . . 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 2838 . . . . . . . . . . 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 462 . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  y  e.  T
)
933, 14lssel 16997 . . . . . . . . . . . . . 14  |-  ( ( T  e.  S  /\  y  e.  T )  ->  y  e.  ( Base `  W ) )
9447, 92, 93syl2anc 656 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  y  e.  (
Base `  W )
)
9594adantr 462 . . . . . . . . . . . 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 458 . . . . . . . . . . . 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 32359 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
)  /\  y  =/=  .0.  )  ->  ( (
LSpan `  W ) `  { y } )  e.  A )
9891, 95, 96, 97syl3anc 1213 . . . . . . . . . . 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 17055 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  ( ( LSpan `  W ) `  {
y } )  C_  T )
10099adantr 462 . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  r  =  ( y ( +g  `  W
) z ) )
102101sneqd 3886 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  { r }  =  { ( y ( +g  `  W
) z ) } )
103102fveq2d 5692 . . . . . . . . . . . . . . 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 17155 . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . 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 3387 . . . . . . . . . . . . . 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 17148 . . . . . . . . . . . . . 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 3389 . . . . . . . . . . . . 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 17036 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { y } )  e.  S )
11144, 94, 110syl2anc 656 . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  r  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  (
y  e.  T  /\  z  e.  U )  /\  r  =  (
y ( +g  `  W
) z ) )  ->  U  e.  (SubGrp `  W ) )
11418lsmless2 16152 . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 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 3363 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 3374 . . . . . . . . . . . . 13  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( p  C_  T  <->  ( ( LSpan `  W ) `  { y } ) 
C_  T ) )
119 oveq1 6097 . . . . . . . . . . . . . 14  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( p  .(+)  U )  =  ( ( (
LSpan `  W ) `  { y } ) 
.(+)  U ) )
120119sseq2d 3381 . . . . . . . . . . . . 13  |-  ( p  =  ( ( LSpan `  W ) `  {
y } )  -> 
( ( ( LSpan `  W ) `  {
r } )  C_  ( p  .(+)  U )  <-> 
( ( LSpan `  W
) `  { r } )  C_  (
( ( LSpan `  W
) `  { y } )  .(+)  U ) ) )
121118, 120anbi12d 705 . . . . . . . . . . . 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 3070 . . . . . . . . . . 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 1211 . . . . . . . . . 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 2687 . . . . . . . . 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 1181 . . . . . . . 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 2845 . . . . . . 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 1003 . . . . . 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 3374 . . . . . . . 8  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( Q  C_  (
p  .(+)  U )  <->  ( ( LSpan `  W ) `  { r } ) 
C_  ( p  .(+)  U ) ) )
130129anbi2d 698 . . . . . . 7  |-  ( Q  =  ( ( LSpan `  W ) `  {
r } )  -> 
( ( p  C_  T  /\  Q  C_  (
p  .(+)  U ) )  <-> 
( p  C_  T  /\  ( ( LSpan `  W
) `  { r } )  C_  (
p  .(+)  U ) ) ) )
131130rexbidv 2734 . . . . . 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 1006 . . . . 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 1181 . . 3  |-  ( ph  ->  ( r  e.  ( ( Base `  W
)  \  {  .0.  } )  ->  ( Q  =  ( ( LSpan `  W ) `  {
r } )  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) ) ) )
135134rexlimdv 2838 . 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 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    \ cdif 3322    C_ wss 3325   {csn 3874   {cpr 3876   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374  SubGrpcsubg 15668   LSSumclsm 16126   LModclmod 16928   LSubSpclss 16991   LSpanclspn 17030  LSAtomsclsa 32341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-cntz 15828  df-lsm 16128  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lss 16992  df-lsp 17031  df-lsatoms 32343
This theorem is referenced by:  dochexmidlem4  34830
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