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Theorem lsat0cv 32683
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 465 . . 3  |-  ( (
ph  /\  U  e.  A )  ->  W  e.  LVec )
6 simpr 461 . . 3  |-  ( (
ph  /\  U  e.  A )  ->  U  e.  A )
71, 2, 3, 5, 6lsatcv0 32681 . 2  |-  ( (
ph  /\  U  e.  A )  ->  {  .0.  } C U )
8 lsat0cv.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
9 lveclmod 17192 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
104, 9syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
1110adantr 465 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  W  e.  LMod )
121, 8lsssn0 17034 . . . . . . . . 9  |-  ( W  e.  LMod  ->  {  .0.  }  e.  S )
1310, 12syl 16 . . . . . . . 8  |-  ( ph  ->  {  .0.  }  e.  S )
1413adantr 465 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  }  e.  S
)
15 lsat0cv.u . . . . . . . 8  |-  ( ph  ->  U  e.  S )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  U  e.  S )
17 simpr 461 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  } C U )
188, 3, 11, 14, 16, 17lcvpss 32674 . . . . . 6  |-  ( (
ph  /\  {  .0.  } C U )  ->  {  .0.  }  C.  U
)
19 pssnel 3749 . . . . . 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 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  U  e.  S )
22 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  U )
23 eqid 2443 . . . . . . . . . . . 12  |-  ( Base `  W )  =  (
Base `  W )
2423, 8lssel 17024 . . . . . . . . . . 11  |-  ( ( U  e.  S  /\  x  e.  U )  ->  x  e.  ( Base `  W ) )
2521, 22, 24syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  ( Base `  W )
)
26 elsn 3896 . . . . . . . . . . . . . 14  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2726biimpri 206 . . . . . . . . . . . . 13  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2827necon3bi 2657 . . . . . . . . . . . 12  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2928adantl 466 . . . . . . . . . . 11  |-  ( ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  x  =/=  .0.  )
3029adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  =/=  .0.  )
31 eldifsn 4005 . . . . . . . . . 10  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  <->  ( x  e.  ( Base `  W
)  /\  x  =/=  .0.  ) )
3225, 30, 31sylanbrc 664 . . . . . . . . 9  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )
3332, 22jca 532 . . . . . . . 8  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  ( x  e.  U  /\  -.  x  e.  {  .0.  } ) )  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) )
3433ex 434 . . . . . . 7  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( ( x  e.  U  /\  -.  x  e.  {  .0.  } )  ->  ( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) ) )
3534eximdv 1676 . . . . . 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 2726 . . . . . 6  |-  ( E. x  e.  ( (
Base `  W )  \  {  .0.  } ) x  e.  U  <->  E. x
( x  e.  ( ( Base `  W
)  \  {  .0.  } )  /\  x  e.  U ) )
3735, 36syl6ibr 227 . . . . 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 758 . . . . . . . 8  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  {  .0.  } C U )
408, 3, 4, 13, 15lcvbr2 32672 . . . . . . . . . . 11  |-  ( ph  ->  ( {  .0.  } C U  <->  ( {  .0.  } 
C.  U  /\  A. s  e.  S  (
( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4140adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  {  .0.  } C U )  -> 
( {  .0.  } C U  <->  ( {  .0.  } 
C.  U  /\  A. s  e.  S  (
( {  .0.  }  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
4241ad2antrr 725 . . . . . . . . 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 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  W  e.  LMod )
4443ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  W  e.  LMod )
45 eldifi 3483 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  e.  ( Base `  W
) )
4645adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  e.  ( Base `  W
) )
4746ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  e.  ( Base `  W ) )
48 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( LSpan `  W )  =  (
LSpan `  W )
4923, 8, 48lspsncl 17063 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
) )  ->  (
( LSpan `  W ) `  { x } )  e.  S )
5044, 47, 49syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W
) `  { x } )  e.  S
)
511, 8lss0ss 17035 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
( LSpan `  W ) `  { x } )  e.  S )  ->  {  .0.  }  C_  (
( LSpan `  W ) `  { x } ) )
5244, 50, 51syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  C_  ( ( LSpan `  W
) `  { x } ) )
53 eldifsni 4006 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( Base `  W )  \  {  .0.  } )  ->  x  =/=  .0.  )
5453adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  x  =/=  .0.  )
5554ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  =/=  .0.  )
5623, 1, 48lspsneq0 17098 . . . . . . . . . . . . . . . . 17  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  W
) )  ->  (
( ( LSpan `  W
) `  { x } )  =  {  .0.  }  <->  x  =  .0.  ) )
5744, 47, 56syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( ( LSpan `  W ) `  {
x } )  =  {  .0.  }  <->  x  =  .0.  ) )
5857necon3bid 2648 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( ( LSpan `  W ) `  {
x } )  =/= 
{  .0.  }  <->  x  =/=  .0.  ) )
5955, 58mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W
) `  { x } )  =/=  {  .0.  } )
6059necomd 2700 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  =/=  ( ( LSpan `  W
) `  { x } ) )
61 df-pss 3349 . . . . . . . . . . . . 13  |-  ( {  .0.  }  C.  (
( LSpan `  W ) `  { x } )  <-> 
( {  .0.  }  C_  ( ( LSpan `  W
) `  { x } )  /\  {  .0.  }  =/=  ( (
LSpan `  W ) `  { x } ) ) )
6252, 60, 61sylanbrc 664 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  {  .0.  }  C.  ( ( LSpan `  W
) `  { x } ) )
6315ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  U  e.  S )
6463ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  U  e.  S )
65 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  x  e.  U )
668, 48, 44, 64, 65lspsnel5a 17082 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
{  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  /\  {  .0.  }  C.  U )  ->  ( ( LSpan `  W
) `  { x } )  C_  U
)
6762, 66jca 532 . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . 15  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( {  .0.  }  C.  s  <->  {  .0.  }  C.  ( ( LSpan `  W
) `  { x } ) ) )
69 sseq1 3382 . . . . . . . . . . . . . . 15  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  C_  U  <->  ( ( LSpan `  W ) `  { x } ) 
C_  U ) )
7068, 69anbi12d 710 . . . . . . . . . . . . . 14  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( ( {  .0.  } 
C.  s  /\  s  C_  U )  <->  ( {  .0.  }  C.  ( ( LSpan `  W ) `  { x } )  /\  ( ( LSpan `  W ) `  {
x } )  C_  U ) ) )
71 eqeq1 2449 . . . . . . . . . . . . . 14  |-  ( s  =  ( ( LSpan `  W ) `  {
x } )  -> 
( s  =  U  <-> 
( ( LSpan `  W
) `  { x } )  =  U ) )
7270, 71imbi12d 320 . . . . . . . . . . . . 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 3074 . . . . . . . . . . . 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 41 . . . . . . . . . 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 603 . . . . . . . . 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 215 . . . . . . . 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 2448 . . . . . 6  |-  ( ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  /\  x  e.  U )  ->  U  =  ( ( LSpan `  W ) `  {
x } ) )
8079ex 434 . . . . 5  |-  ( ( ( ph  /\  {  .0.  } C U )  /\  x  e.  ( ( Base `  W
)  \  {  .0.  } ) )  ->  (
x  e.  U  ->  U  =  ( ( LSpan `  W ) `  { x } ) ) )
8180reximdva 2833 . . . 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 465 . . . 4  |-  ( (
ph  /\  {  .0.  } C U )  ->  W  e.  LVec )
8423, 48, 1, 2islsat 32641 . . . 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 232 . 2  |-  ( (
ph  /\  {  .0.  } C U )  ->  U  e.  A )
877, 86impbida 828 1  |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721    \ cdif 3330    C_ wss 3333    C. wpss 3334   {csn 3882   class class class wbr 4297   ` cfv 5423   Basecbs 14179   0gc0g 14383   LModclmod 16953   LSubSpclss 17018   LSpanclspn 17057   LVecclvec 17188  LSAtomsclsa 32624    <oLL clcv 32668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-drng 16839  df-lmod 16955  df-lss 17019  df-lsp 17058  df-lvec 17189  df-lsatoms 32626  df-lcv 32669
This theorem is referenced by:  mapdcnvatN  35316  mapdat  35317
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