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Theorem lsat0cv 29516
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 452 . . 3  |-  ( (
ph  /\  U  e.  A )  ->  W  e.  LVec )
6 simpr 448 . . 3  |-  ( (
ph  /\  U  e.  A )  ->  U  e.  A )
71, 2, 3, 5, 6lsatcv0 29514 . 2  |-  ( (
ph  /\  U  e.  A )  ->  {  .0.  } C U )
8 lsat0cv.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
9 lveclmod 16133 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
104, 9syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
1110adantr 452 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  W  e.  LMod )
121, 8lsssn0 15979 . . . . . . . . 9  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )
1310, 12syl 16 . . . . . . . 8  |-  ( ph  ->  {  .0.  }  e.  S )
1413adantr 452 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  }  e.  S
)
15 lsat0cv.u . . . . . . . 8  |-  ( ph  ->  U  e.  S )
1615adantr 452 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  U  e.  S )
17 simpr 448 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  } C U )
188, 3, 11, 14, 16, 17lcvpss 29507 . . . . . 6  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  }  C.  U
)
19 pssnel 3653 . . . . . 6  |-  ( {  .0.  }  C.  U  ->  E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )
2018, 19syl 16 . . . . 5  |-  ( (
ph  /\  {  .0.  } C U )  ->  E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )
2115ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  U  e.  S )
22 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  U )
23 eqid 2404 . . . . . . . . . . . 12  |-  ( Base `  W )  =  (
Base `  W )
2423, 8lssel 15969 . . . . . . . . . . 11  |-  ( ( U  e.  S  /\  x  e.  U )  ->  x  e.  ( Base `  W ) )
2521, 22, 24syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  ( Base `  W )
)
26 elsn 3789 . . . . . . . . . . . . . 14  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2726biimpri 198 . . . . . . . . . . . . 13  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2827necon3bi 2608 . . . . . . . . . . . 12  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2928adantl 453 . . . . . . . . . . 11  |-  ( ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  x  =/=  .0.  )
3029adantl 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  =/=  .0.  )
31 eldifsn 3887 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( x  e.  ( Base `  W
)  /\  x  =/=  .0.  ) )
3225, 30, 31sylanbrc 646 . . . . . . . . 9  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )
3332, 22jca 519 . . . . . . . 8  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) )
3433ex 424 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) ) )
3534eximdv 1629 . . . . . 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 2672 . . . . . 6  |-  ( E. x  e.  ( (
Base `  W )  \  {  .0.  } ) x  e.  U  <->  E. x
( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) )
3735, 36syl6ibr 219 . . . . 5  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( E. x ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) x  e.  U
) )
3820, 37mpd 15 . . . 4  |-  ( (
ph  /\  {  .0.  } C U )  ->  E. x  e.  (
( Base `  W )  \  {  .0.  } ) x  e.  U )
39 simpllr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  {  .0.  } C U )
408, 3, 4, 13, 15lcvbr2 29505 . . . . . . . . . . 11  |-  ( ph  ->  ( {  .0.  } C U  <->  ( {  .0.  } 
C.  U  /\  A. s  e.  S  (
( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4140adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( {  .0.  } C U  <->  ( {  .0.  } 
C.  U  /\  A. s  e.  S  (
( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4241ad2antrr 707 . . . . . . . . 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 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LMod )
4443ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  W  e.  LMod )
45 eldifi 3429 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  e.  ( Base `  W
) )
4645adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  e.  ( Base `  W
) )
4746ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  e.  (
Base `  W )
)
48 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( LSpan `  W )  =  (
LSpan `  W )
4923, 8, 48lspsncl 16008 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { x } )  e.  S )
5044, 47, 49syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W ) `  {
x } )  e.  S )
511, 8lss0ss 15980 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
( LSpan `  W ) `  { x } )  e.  S )  ->  {  .0.  }  C_  (
( LSpan `  W ) `  { x } ) )
5244, 50, 51syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  C_  ( ( LSpan `  W
) `  { x } ) )
53 eldifsni 3888 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  =/=  .0.  )
5453adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  =/=  .0.  )
5554ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  =/=  .0.  )
5623, 1, 48lspsneq0 16043 . . . . . . . . . . . . . . . . 17  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
) )  ->  (
( ( LSpan `  W
) `  { x } )  =  {  .0.  }  <->  x  =  .0.  ) )
5744, 47, 56syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( (
LSpan `  W ) `  { x } )  =  {  .0.  }  <->  x  =  .0.  ) )
5857necon3bid 2602 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( (
LSpan `  W ) `  { x } )  =/=  {  .0.  }  <->  x  =/=  .0.  ) )
5955, 58mpbird 224 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W ) `  {
x } )  =/= 
{  .0.  } )
6059necomd 2650 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  =/=  ( ( LSpan `  W
) `  { x } ) )
61 df-pss 3296 . . . . . . . . . . . . 13  |-  ( {  .0.  }  C.  (
( LSpan `  W ) `  { x } )  <-> 
( {  .0.  }  C_  ( ( LSpan `  W
) `  { x } )  /\  {  .0.  }  =/=  ( (
LSpan `  W ) `  { x } ) ) )
6252, 60, 61sylanbrc 646 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  C.  ( ( LSpan `  W
) `  { x } ) )
6315ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  U  e.  S )
6463ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  U  e.  S
)
65 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  e.  U
)
668, 48, 44, 64, 65lspsnel5a 16027 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W ) `  {
x } )  C_  U )
6762, 66jca 519 . . . . . . . . . . 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 3395 . . . . . . . . . . . . . . 15  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( {  .0.  }  C.  s  <->  {  .0.  }  C.  ( ( LSpan `  W
) `  { x } ) ) )
69 sseq1 3329 . . . . . . . . . . . . . . 15  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
7068, 69anbi12d 692 . . . . . . . . . . . . . 14  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( {  .0.  } 
C.  s  /\  s  C_  U )  <->  ( {  .0.  }  C.  ( (
LSpan `  W ) `  { x } )  /\  ( ( LSpan `  W ) `  {
x } )  C_  U ) ) )
71 eqeq1 2410 . . . . . . . . . . . . . 14  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  =  U  <-> 
( ( LSpan `  W
) `  { x } )  =  U ) )
7270, 71imbi12d 312 . . . . . . . . . . . . 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 3008 . . . . . . . . . . . 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 16 . . . . . . . . . . 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 39 . . . . . . . . . 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 587 . . . . . . . . 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 207 . . . . . . . 8  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  ( {  .0.  } C U  ->  ( ( LSpan `  W ) `  {
x } )  =  U ) )
7839, 77mpd 15 . . . . . . 7  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  (
( LSpan `  W ) `  { x } )  =  U )
7978eqcomd 2409 . . . . . 6  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  U  =  ( ( LSpan `  W ) `  {
x } ) )
8079ex 424 . . . . 5  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  (
x  e.  U  ->  U  =  ( ( LSpan `  W ) `  { x } ) ) )
8180reximdva 2778 . . . 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 15 . . 3  |-  ( (
ph  /\  {  .0.  } C U )  ->  E. x  e.  (
( Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { x } ) )
834adantr 452 . . . 4  |-  ( (
ph  /\  {  .0.  } C U )  ->  W  e.  LVec )
8423, 48, 1, 2islsat 29474 . . . 4  |-  ( W  e.  LVec  ->  ( U  e.  A  <->  E. x  e.  ( ( Base `  W
)  \  {  .0.  } ) U  =  ( ( LSpan `  W ) `  { x } ) ) )
8583, 84syl 16 . . 3  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( U  e.  A  <->  E. x  e.  ( (
Base `  W )  \  {  .0.  } ) U  =  ( (
LSpan `  W ) `  { x } ) ) )
8682, 85mpbird 224 . 2  |-  ( (
ph  /\  {  .0.  } C U )  ->  U  e.  A )
877, 86impbida 806 1  |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277    C_ wss 3280    C. wpss 3281   {csn 3774   class class class wbr 4172   ` cfv 5413   Basecbs 13424   0gc0g 13678   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   LVecclvec 16129  LSAtomsclsa 29457    <oLL clcv 29501
This theorem is referenced by:  mapdcnvatN  32149  mapdat  32150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lcv 29502
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