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Theorem lsat0cv 29223
Description: A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
Hypotheses
Ref Expression
lsat0cv.o  |-  .0.  =  ( 0g `  W )
lsat0cv.s  |-  S  =  ( LSubSp `  W )
lsat0cv.a  |-  A  =  (LSAtoms `  W )
lsat0cv.c  |-  C  =  (  <oLL  `  W )
lsat0cv.w  |-  ( ph  ->  W  e.  LVec )
lsat0cv.u  |-  ( ph  ->  U  e.  S )
Assertion
Ref Expression
lsat0cv  |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )

Proof of Theorem lsat0cv
Dummy variables  x  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsat0cv.o . . 3  |-  .0.  =  ( 0g `  W )
2 lsat0cv.a . . 3  |-  A  =  (LSAtoms `  W )
3 lsat0cv.c . . 3  |-  C  =  (  <oLL  `  W )
4 lsat0cv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
54adantr 451 . . 3  |-  ( (
ph  /\  U  e.  A )  ->  W  e.  LVec )
6 simpr 447 . . 3  |-  ( (
ph  /\  U  e.  A )  ->  U  e.  A )
71, 2, 3, 5, 6lsatcv0 29221 . 2  |-  ( (
ph  /\  U  e.  A )  ->  {  .0.  } C U )
8 lsat0cv.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
9 lveclmod 15859 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
104, 9syl 15 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
1110adantr 451 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  W  e.  LMod )
121, 8lsssn0 15705 . . . . . . . . 9  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )
1310, 12syl 15 . . . . . . . 8  |-  ( ph  ->  {  .0.  }  e.  S )
1413adantr 451 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  }  e.  S
)
15 lsat0cv.u . . . . . . . 8  |-  ( ph  ->  U  e.  S )
1615adantr 451 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  U  e.  S )
17 simpr 447 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  } C U )
188, 3, 11, 14, 16, 17lcvpss 29214 . . . . . 6  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  }  C.  U
)
19 pssnel 3519 . . . . . 6  |-  ( {  .0.  }  C.  U  ->  E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )
2018, 19syl 15 . . . . 5  |-  ( (
ph  /\  {  .0.  } C U )  ->  E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )
2115ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  U  e.  S )
22 simprl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  U )
23 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  W )  =  (
Base `  W )
2423, 8lssel 15695 . . . . . . . . . . 11  |-  ( ( U  e.  S  /\  x  e.  U )  ->  x  e.  ( Base `  W ) )
2521, 22, 24syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  ( Base `  W )
)
26 elsn 3655 . . . . . . . . . . . . . 14  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2726biimpri 197 . . . . . . . . . . . . 13  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2827necon3bi 2487 . . . . . . . . . . . 12  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2928adantl 452 . . . . . . . . . . 11  |-  ( ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  x  =/=  .0.  )
3029adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  =/=  .0.  )
31 eldifsn 3749 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( x  e.  ( Base `  W
)  /\  x  =/=  .0.  ) )
3225, 30, 31sylanbrc 645 . . . . . . . . 9  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )
3332, 22jca 518 . . . . . . . 8  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) )
3433ex 423 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) ) )
3534eximdv 1608 . . . . . 6  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  E. x
( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) ) )
36 df-rex 2549 . . . . . 6  |-  ( E. x  e.  ( (
Base `  W )  \  {  .0.  } ) x  e.  U  <->  E. x
( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) )
3735, 36syl6ibr 218 . . . . 5  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) x  e.  U
) )
3820, 37mpd 14 . . . 4  |-  ( (
ph  /\  {  .0.  } C U )  ->  E. x  e.  (
( Base `  W )  \  {  .0.  } ) x  e.  U )
39 simpllr 735 . . . . . . . 8  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  {  .0.  } C U )
408, 3, 4, 13, 15lcvbr2 29212 . . . . . . . . . . 11  |-  ( ph  ->  ( {  .0.  } C U  <->  ( {  .0.  } 
C.  U  /\  A. s  e.  S  (
( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4140adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( {  .0.  } C U  <->  ( {  .0.  } 
C.  U  /\  A. s  e.  S  (
( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4241ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  ( {  .0.  } C U  <-> 
( {  .0.  }  C.  U  /\  A. s  e.  S  ( ( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4310ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LMod )
4443ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  W  e.  LMod )
45 eldifi 3298 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  e.  ( Base `  W
) )
4645adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  e.  ( Base `  W
) )
4746ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  e.  (
Base `  W )
)
48 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( LSpan `  W )  =  (
LSpan `  W )
4923, 8, 48lspsncl 15734 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { x } )  e.  S )
5044, 47, 49syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W ) `  {
x } )  e.  S )
511, 8lss0ss 15706 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
( LSpan `  W ) `  { x } )  e.  S )  ->  {  .0.  }  C_  (
( LSpan `  W ) `  { x } ) )
5244, 50, 51syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  C_  ( ( LSpan `  W
) `  { x } ) )
53 eldifsni 3750 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  =/=  .0.  )
5453adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  =/=  .0.  )
5554ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  =/=  .0.  )
5623, 1, 48lspsneq0 15769 . . . . . . . . . . . . . . . . 17  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
) )  ->  (
( ( LSpan `  W
) `  { x } )  =  {  .0.  }  <->  x  =  .0.  ) )
5744, 47, 56syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( (
LSpan `  W ) `  { x } )  =  {  .0.  }  <->  x  =  .0.  ) )
5857necon3bid 2481 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( (
LSpan `  W ) `  { x } )  =/=  {  .0.  }  <->  x  =/=  .0.  ) )
5955, 58mpbird 223 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W ) `  {
x } )  =/= 
{  .0.  } )
6059necomd 2529 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  =/=  ( ( LSpan `  W
) `  { x } ) )
61 df-pss 3168 . . . . . . . . . . . . 13  |-  ( {  .0.  }  C.  (
( LSpan `  W ) `  { x } )  <-> 
( {  .0.  }  C_  ( ( LSpan `  W
) `  { x } )  /\  {  .0.  }  =/=  ( (
LSpan `  W ) `  { x } ) ) )
6252, 60, 61sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  C.  ( ( LSpan `  W
) `  { x } ) )
6315ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  U  e.  S )
6463ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  U  e.  S
)
65 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  e.  U
)
668, 48, 44, 64, 65lspsnel5a 15753 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
6762, 66jca 518 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( {  .0.  } 
C.  ( ( LSpan `  W ) `  {
x } )  /\  ( ( LSpan `  W
) `  { x } )  C_  U
) )
68 psseq2 3264 . . . . . . . . . . . . . . 15  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( {  .0.  }  C.  s  <->  {  .0.  }  C.  ( ( LSpan `  W
) `  { x } ) ) )
69 sseq1 3199 . . . . . . . . . . . . . . 15  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
7068, 69anbi12d 691 . . . . . . . . . . . . . 14  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( {  .0.  } 
C.  s  /\  s  C_  U )  <->  ( {  .0.  }  C.  ( (
LSpan `  W ) `  { x } )  /\  ( ( LSpan `  W ) `  {
x } )  C_  U ) ) )
71 eqeq1 2289 . . . . . . . . . . . . . 14  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  =  U  <-> 
( ( LSpan `  W
) `  { x } )  =  U ) )
7270, 71imbi12d 311 . . . . . . . . . . . . 13  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( ( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  ( ( {  .0.  }  C.  (
( LSpan `  W ) `  { x } )  /\  ( ( LSpan `  W ) `  {
x } )  C_  U )  ->  (
( LSpan `  W ) `  { x } )  =  U ) ) )
7372rspcv 2880 . . . . . . . . . . . 12  |-  ( ( ( LSpan `  W ) `  { x } )  e.  S  ->  ( A. s  e.  S  ( ( {  .0.  } 
C.  s  /\  s  C_  U )  ->  s  =  U )  ->  (
( {  .0.  }  C.  ( ( LSpan `  W
) `  { x } )  /\  (
( LSpan `  W ) `  { x } ) 
C_  U )  -> 
( ( LSpan `  W
) `  { x } )  =  U ) ) )
7450, 73syl 15 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( A. s  e.  S  ( ( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U )  ->  ( ( {  .0.  }  C.  (
( LSpan `  W ) `  { x } )  /\  ( ( LSpan `  W ) `  {
x } )  C_  U )  ->  (
( LSpan `  W ) `  { x } )  =  U ) ) )
7567, 74mpid 37 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( A. s  e.  S  ( ( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U )  ->  ( ( LSpan `  W ) `  { x } )  =  U ) )
7675expimpd 586 . . . . . . . . 9  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  (
( {  .0.  }  C.  U  /\  A. s  e.  S  ( ( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) )  ->  (
( LSpan `  W ) `  { x } )  =  U ) )
7742, 76sylbid 206 . . . . . . . 8  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  ( {  .0.  } C U  ->  ( ( LSpan `  W ) `  {
x } )  =  U ) )
7839, 77mpd 14 . . . . . . 7  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  (
( LSpan `  W ) `  { x } )  =  U )
7978eqcomd 2288 . . . . . 6  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  U  =  ( ( LSpan `  W ) `  {
x } ) )
8079ex 423 . . . . 5  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  (
x  e.  U  ->  U  =  ( ( LSpan `  W ) `  { x } ) ) )
8180reximdva 2655 . . . 4  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) x  e.  U  ->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { x } ) ) )
8238, 81mpd 14 . . 3  |-  ( (
ph  /\  {  .0.  } C U )  ->  E. x  e.  (
( Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { x } ) )
834adantr 451 . . . 4  |-  ( (
ph  /\  {  .0.  } C U )  ->  W  e.  LVec )
8423, 48, 1, 2islsat 29181 . . . 4  |-  ( W  e.  LVec  ->  ( U  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { x } ) ) )
8583, 84syl 15 . . 3  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( U  e.  A  <->  E. x  e.  ( (
Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { x } ) ) )
8682, 85mpbird 223 . 2  |-  ( (
ph  /\  {  .0.  } C U )  ->  U  e.  A )
877, 86impbida 805 1  |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152    C. wpss 3153   {csn 3640   class class class wbr 4023   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSAtomsclsa 29164    <oLL clcv 29208
This theorem is referenced by:  mapdcnvatN  31856  mapdat  31857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lcv 29209
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