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Theorem djhcvat42 31910
Description: A covering property. (cvrat42 29938 analog.) (Contributed by NM, 17-Aug-2014.)
Hypotheses
Ref Expression
djhcvat42.h  |-  H  =  ( LHyp `  K
)
djhcvat42.u  |-  U  =  ( ( DVecH `  K
) `  W )
djhcvat42.v  |-  V  =  ( Base `  U
)
djhcvat42.o  |-  .0.  =  ( 0g `  U )
djhcvat42.n  |-  N  =  ( LSpan `  U )
djhcvat42.i  |-  I  =  ( ( DIsoH `  K
) `  W )
djhcvat42.j  |-  .\/  =  ( (joinH `  K ) `  W )
djhcvat42.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
djhcvat42.s  |-  ( ph  ->  S  e.  ran  I
)
djhcvat42.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
djhcvat42.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
djhcvat42  |-  ( ph  ->  ( ( S  =/= 
{  .0.  }  /\  ( N `  { X } )  C_  ( S  .\/  ( N `  { Y } ) ) )  ->  E. z  e.  ( V  \  {  .0.  } ) ( ( N `  { z } )  C_  S  /\  ( N `  { X } )  C_  (
( N `  {
z } )  .\/  ( N `  { Y } ) ) ) ) )
Distinct variable groups:    z, I    z, K    z, N    ph, z    z, W    z, S    z, V    z, X    z, Y
Allowed substitution hints:    U( z)    H( z)    .\/ ( z)    .0. ( z)

Proof of Theorem djhcvat42
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 djhcvat42.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
21simpld 446 . . 3  |-  ( ph  ->  K  e.  HL )
3 djhcvat42.s . . . 4  |-  ( ph  ->  S  e.  ran  I
)
4 eqid 2412 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
5 djhcvat42.h . . . . 5  |-  H  =  ( LHyp `  K
)
6 djhcvat42.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
74, 5, 6dihcnvcl 31766 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  ran  I )  ->  ( `' I `  S )  e.  ( Base `  K
) )
81, 3, 7syl2anc 643 . . 3  |-  ( ph  ->  ( `' I `  S )  e.  (
Base `  K )
)
9 djhcvat42.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
109eldifad 3300 . . . 4  |-  ( ph  ->  X  e.  V )
11 eldifsni 3896 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
129, 11syl 16 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
13 eqid 2412 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
14 djhcvat42.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
15 djhcvat42.v . . . . 5  |-  V  =  ( Base `  U
)
16 djhcvat42.o . . . . 5  |-  .0.  =  ( 0g `  U )
17 djhcvat42.n . . . . 5  |-  N  =  ( LSpan `  U )
1813, 5, 14, 15, 16, 17, 6dihlspsnat 31828 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )
)
191, 10, 12, 18syl3anc 1184 . . 3  |-  ( ph  ->  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )
)
20 djhcvat42.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3300 . . . 4  |-  ( ph  ->  Y  e.  V )
22 eldifsni 3896 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2320, 22syl 16 . . . 4  |-  ( ph  ->  Y  =/=  .0.  )
2413, 5, 14, 15, 16, 17, 6dihlspsnat 31828 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V  /\  Y  =/=  .0.  )  ->  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
)
251, 21, 23, 24syl3anc 1184 . . 3  |-  ( ph  ->  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
)
26 eqid 2412 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
27 eqid 2412 . . . 4  |-  ( join `  K )  =  (
join `  K )
28 eqid 2412 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
294, 26, 27, 28, 13cvrat42 29938 . . 3  |-  ( ( K  e.  HL  /\  ( ( `' I `  S )  e.  (
Base `  K )  /\  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )  /\  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
) )  ->  (
( ( `' I `  S )  =/=  ( 0. `  K )  /\  ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( ( `' I `  S ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )  ->  E. r  e.  (
Atoms `  K ) ( r ( le `  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( r ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) ) ) )
302, 8, 19, 25, 29syl13anc 1186 . 2  |-  ( ph  ->  ( ( ( `' I `  S )  =/=  ( 0. `  K )  /\  ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( ( `' I `  S ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )  ->  E. r  e.  (
Atoms `  K ) ( r ( le `  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( r ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) ) ) )
315, 28, 6, 14, 15, 16, 17, 1, 3dih0sb 31780 . . . 4  |-  ( ph  ->  ( S  =  {  .0.  }  <->  ( `' I `  S )  =  ( 0. `  K ) ) )
3231necon3bid 2610 . . 3  |-  ( ph  ->  ( S  =/=  {  .0.  }  <->  ( `' I `  S )  =/=  ( 0. `  K ) ) )
335, 14, 15, 17, 6dihlsprn 31826 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( N `  { X } )  e.  ran  I )
341, 10, 33syl2anc 643 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  ran  I )
355, 14, 6, 15dihrnss 31773 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  ran  I )  ->  S  C_  V )
361, 3, 35syl2anc 643 . . . . . 6  |-  ( ph  ->  S  C_  V )
375, 14, 15, 17, 6dihlsprn 31826 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( N `  { Y } )  e.  ran  I )
381, 21, 37syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  e.  ran  I )
395, 14, 6, 15dihrnss 31773 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { Y } )  e. 
ran  I )  -> 
( N `  { Y } )  C_  V
)
401, 38, 39syl2anc 643 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
41 djhcvat42.j . . . . . . 7  |-  .\/  =  ( (joinH `  K ) `  W )
425, 6, 14, 15, 41djhcl 31895 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  V  /\  ( N `  { Y } )  C_  V ) )  -> 
( S  .\/  ( N `  { Y } ) )  e. 
ran  I )
431, 36, 40, 42syl12anc 1182 . . . . 5  |-  ( ph  ->  ( S  .\/  ( N `  { Y } ) )  e. 
ran  I )
4426, 5, 6, 1, 34, 43dihcnvord 31769 . . . 4  |-  ( ph  ->  ( ( `' I `  ( N `  { X } ) ) ( le `  K ) ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  <->  ( N `  { X } ) 
C_  ( S  .\/  ( N `  { Y } ) ) ) )
4527, 5, 6, 41, 1, 3, 38djhj 31899 . . . . 5  |-  ( ph  ->  ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  =  ( ( `' I `  S ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
4645breq2d 4192 . . . 4  |-  ( ph  ->  ( ( `' I `  ( N `  { X } ) ) ( le `  K ) ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  <->  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  S ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) )
4744, 46bitr3d 247 . . 3  |-  ( ph  ->  ( ( N `  { X } )  C_  ( S  .\/  ( N `
 { Y }
) )  <->  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  S ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) )
4832, 47anbi12d 692 . 2  |-  ( ph  ->  ( ( S  =/= 
{  .0.  }  /\  ( N `  { X } )  C_  ( S  .\/  ( N `  { Y } ) ) )  <->  ( ( `' I `  S )  =/=  ( 0. `  K )  /\  ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( ( `' I `  S ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) ) ) )
491adantr 452 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
50 eldifi 3437 . . . . . 6  |-  ( z  e.  ( V  \  {  .0.  } )  -> 
z  e.  V )
5150adantl 453 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  z  e.  V )
52 eldifsni 3896 . . . . . 6  |-  ( z  e.  ( V  \  {  .0.  } )  -> 
z  =/=  .0.  )
5352adantl 453 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  z  =/=  .0.  )
5413, 5, 14, 15, 16, 17, 6dihlspsnat 31828 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  V  /\  z  =/=  .0.  )  ->  ( `' I `  ( N `  {
z } ) )  e.  ( Atoms `  K
) )
5549, 51, 53, 54syl3anc 1184 . . . 4  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( N `
 { z } ) )  e.  (
Atoms `  K ) )
5613, 5, 14, 15, 16, 17, 6, 1dihatexv2 31834 . . . 4  |-  ( ph  ->  ( r  e.  (
Atoms `  K )  <->  E. z  e.  ( V  \  {  .0.  } ) r  =  ( `' I `  ( N `  { z } ) ) ) )
57 breq1 4183 . . . . . 6  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( r ( le `  K ) ( `' I `  S )  <->  ( `' I `  ( N `  { z } ) ) ( le `  K ) ( `' I `  S ) ) )
58 oveq1 6055 . . . . . . 7  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( r (
join `  K )
( `' I `  ( N `  { Y } ) ) )  =  ( ( `' I `  ( N `
 { z } ) ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
5958breq2d 4192 . . . . . 6  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( r ( join `  K
) ( `' I `  ( N `  { Y } ) ) )  <-> 
( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) )
6057, 59anbi12d 692 . . . . 5  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( ( r ( le `  K
) ( `' I `  S )  /\  ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( r ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) )  <->  ( ( `' I `  ( N `
 { z } ) ) ( le
`  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) ) )
6160adantl 453 . . . 4  |-  ( (
ph  /\  r  =  ( `' I `  ( N `
 { z } ) ) )  -> 
( ( r ( le `  K ) ( `' I `  S )  /\  ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( r ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) )  <->  ( ( `' I `  ( N `
 { z } ) ) ( le
`  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) ) )
6255, 56, 61rexxfr2d 4707 . . 3  |-  ( ph  ->  ( E. r  e.  ( Atoms `  K )
( r ( le
`  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( r ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )  <->  E. z  e.  ( V  \  {  .0.  }
) ( ( `' I `  ( N `
 { z } ) ) ( le
`  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) ) )
635, 14, 15, 17, 6dihlsprn 31826 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  V
)  ->  ( N `  { z } )  e.  ran  I )
6449, 51, 63syl2anc 643 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { z } )  e.  ran  I )
653adantr 452 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  S  e.  ran  I )
6626, 5, 6, 49, 64, 65dihcnvord 31769 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( `' I `  ( N `  { z } ) ) ( le `  K ) ( `' I `  S )  <->  ( N `  { z } ) 
C_  S ) )
6738adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { Y } )  e.  ran  I )
6827, 5, 6, 41, 49, 64, 67djhj 31899 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( ( N `  { z } )  .\/  ( N `  { Y } ) ) )  =  ( ( `' I `  ( N `
 { z } ) ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
6968breq2d 4192 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( `' I `  ( N `  { X } ) ) ( le `  K ) ( `' I `  ( ( N `  { z } ) 
.\/  ( N `  { Y } ) ) )  <->  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) )
7010adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  X  e.  V )
7149, 70, 33syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { X } )  e.  ran  I )
725, 14, 6, 15dihrnss 31773 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { z } )  e.  ran  I )  ->  ( N `  { z } ) 
C_  V )
7349, 64, 72syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { z } )  C_  V
)
7440adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { Y } )  C_  V
)
755, 6, 14, 15, 41djhcl 31895 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( N `
 { z } )  C_  V  /\  ( N `  { Y } )  C_  V
) )  ->  (
( N `  {
z } )  .\/  ( N `  { Y } ) )  e. 
ran  I )
7649, 73, 74, 75syl12anc 1182 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( N `  {
z } )  .\/  ( N `  { Y } ) )  e. 
ran  I )
7726, 5, 6, 49, 71, 76dihcnvord 31769 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( `' I `  ( N `  { X } ) ) ( le `  K ) ( `' I `  ( ( N `  { z } ) 
.\/  ( N `  { Y } ) ) )  <->  ( N `  { X } )  C_  ( ( N `  { z } ) 
.\/  ( N `  { Y } ) ) ) )
7869, 77bitr3d 247 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) )  <-> 
( N `  { X } )  C_  (
( N `  {
z } )  .\/  ( N `  { Y } ) ) ) )
7966, 78anbi12d 692 . . . 4  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( ( `' I `  ( N `  {
z } ) ) ( le `  K
) ( `' I `  S )  /\  ( `' I `  ( N `
 { X }
) ) ( le
`  K ) ( ( `' I `  ( N `  { z } ) ) (
join `  K )
( `' I `  ( N `  { Y } ) ) ) )  <->  ( ( N `
 { z } )  C_  S  /\  ( N `  { X } )  C_  (
( N `  {
z } )  .\/  ( N `  { Y } ) ) ) ) )
8079rexbidva 2691 . . 3  |-  ( ph  ->  ( E. z  e.  ( V  \  {  .0.  } ) ( ( `' I `  ( N `
 { z } ) ) ( le
`  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  ( N `  {
z } ) ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) )  <->  E. z  e.  ( V  \  {  .0.  } ) ( ( N `
 { z } )  C_  S  /\  ( N `  { X } )  C_  (
( N `  {
z } )  .\/  ( N `  { Y } ) ) ) ) )
8162, 80bitr2d 246 . 2  |-  ( ph  ->  ( E. z  e.  ( V  \  {  .0.  } ) ( ( N `  { z } )  C_  S  /\  ( N `  { X } )  C_  (
( N `  {
z } )  .\/  ( N `  { Y } ) ) )  <->  E. r  e.  ( Atoms `  K ) ( r ( le `  K ) ( `' I `  S )  /\  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( r ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) ) ) )
8230, 48, 813imtr4d 260 1  |-  ( ph  ->  ( ( S  =/= 
{  .0.  }  /\  ( N `  { X } )  C_  ( S  .\/  ( N `  { Y } ) ) )  ->  E. z  e.  ( V  \  {  .0.  } ) ( ( N `  { z } )  C_  S  /\  ( N `  { X } )  C_  (
( N `  {
z } )  .\/  ( N `  { Y } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    \ cdif 3285    C_ wss 3288   {csn 3782   class class class wbr 4180   `'ccnv 4844   ran crn 4846   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   0gc0g 13686   joincjn 14364   0.cp0 14429   LSpanclspn 16010   Atomscatm 29758   HLchlt 29845   LHypclh 30478   DVecHcdvh 31573   DIsoHcdih 31723  joinHcdjh 31889
This theorem is referenced by:  dihjat1lem  31923
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-tpos 6446  df-undef 6510  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-0g 13690  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-lsm 15233  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-dvr 15751  df-drng 15800  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lvec 16138  df-lsatoms 29471  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tendo 31249  df-edring 31251  df-disoa 31524  df-dvech 31574  df-dib 31634  df-dic 31668  df-dih 31724  df-doch 31843  df-djh 31890
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