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Theorem djhcvat42 37558
Description: A covering property. (cvrat42 35584 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 457 . . 3  |-  ( ph  ->  K  e.  HL )
3 djhcvat42.s . . . 4  |-  ( ph  ->  S  e.  ran  I
)
4 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
5 djhcvat42.h . . . . 5  |-  H  =  ( LHyp `  K
)
6 djhcvat42.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
74, 5, 6dihcnvcl 37414 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  ran  I )  ->  ( `' I `  S )  e.  ( Base `  K
) )
81, 3, 7syl2anc 659 . . 3  |-  ( ph  ->  ( `' I `  S )  e.  (
Base `  K )
)
9 djhcvat42.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
109eldifad 3473 . . . 4  |-  ( ph  ->  X  e.  V )
11 eldifsni 4142 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
129, 11syl 16 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
13 eqid 2454 . . . . 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 37476 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )
)
191, 10, 12, 18syl3anc 1226 . . 3  |-  ( ph  ->  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )
)
20 djhcvat42.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3473 . . . 4  |-  ( ph  ->  Y  e.  V )
22 eldifsni 4142 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2320, 22syl 16 . . . 4  |-  ( ph  ->  Y  =/=  .0.  )
2413, 5, 14, 15, 16, 17, 6dihlspsnat 37476 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V  /\  Y  =/=  .0.  )  ->  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
)
251, 21, 23, 24syl3anc 1226 . . 3  |-  ( ph  ->  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
)
26 eqid 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
27 eqid 2454 . . . 4  |-  ( join `  K )  =  (
join `  K )
28 eqid 2454 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
294, 26, 27, 28, 13cvrat42 35584 . . 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 1228 . 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 37428 . . . 4  |-  ( ph  ->  ( S  =  {  .0.  }  <->  ( `' I `  S )  =  ( 0. `  K ) ) )
3231necon3bid 2712 . . 3  |-  ( ph  ->  ( S  =/=  {  .0.  }  <->  ( `' I `  S )  =/=  ( 0. `  K ) ) )
335, 14, 15, 17, 6dihlsprn 37474 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( N `  { X } )  e.  ran  I )
341, 10, 33syl2anc 659 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  ran  I )
355, 14, 6, 15dihrnss 37421 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  ran  I )  ->  S  C_  V )
361, 3, 35syl2anc 659 . . . . . 6  |-  ( ph  ->  S  C_  V )
375, 14, 15, 17, 6dihlsprn 37474 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( N `  { Y } )  e.  ran  I )
381, 21, 37syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  e.  ran  I )
395, 14, 6, 15dihrnss 37421 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { Y } )  e. 
ran  I )  -> 
( N `  { Y } )  C_  V
)
401, 38, 39syl2anc 659 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
41 djhcvat42.j . . . . . . 7  |-  .\/  =  ( (joinH `  K ) `  W )
425, 6, 14, 15, 41djhcl 37543 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  V  /\  ( N `  { Y } )  C_  V ) )  -> 
( S  .\/  ( N `  { Y } ) )  e. 
ran  I )
431, 36, 40, 42syl12anc 1224 . . . . 5  |-  ( ph  ->  ( S  .\/  ( N `  { Y } ) )  e. 
ran  I )
4426, 5, 6, 1, 34, 43dihcnvord 37417 . . . 4  |-  ( ph  ->  ( ( `' I `  ( N `  { X } ) ) ( le `  K ) ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  <->  ( N `  { X } ) 
C_  ( S  .\/  ( N `  { Y } ) ) ) )
4527, 5, 6, 41, 1, 3, 38djhj 37547 . . . . 5  |-  ( ph  ->  ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  =  ( ( `' I `  S ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
4645breq2d 4451 . . . 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 255 . . 3  |-  ( ph  ->  ( ( N `  { X } )  C_  ( S  .\/  ( N `
 { Y }
) )  <->  ( `' I `  ( N `  { X } ) ) ( le `  K ) ( ( `' I `  S ) ( join `  K
) ( `' I `  ( N `  { Y } ) ) ) ) )
4832, 47anbi12d 708 . 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 463 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
50 eldifi 3612 . . . . . 6  |-  ( z  e.  ( V  \  {  .0.  } )  -> 
z  e.  V )
5150adantl 464 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  z  e.  V )
52 eldifsni 4142 . . . . . 6  |-  ( z  e.  ( V  \  {  .0.  } )  -> 
z  =/=  .0.  )
5352adantl 464 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  z  =/=  .0.  )
5413, 5, 14, 15, 16, 17, 6dihlspsnat 37476 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  V  /\  z  =/=  .0.  )  ->  ( `' I `  ( N `  {
z } ) )  e.  ( Atoms `  K
) )
5549, 51, 53, 54syl3anc 1226 . . . 4  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( N `
 { z } ) )  e.  (
Atoms `  K ) )
5613, 5, 14, 15, 16, 17, 6, 1dihatexv2 37482 . . . 4  |-  ( ph  ->  ( r  e.  (
Atoms `  K )  <->  E. z  e.  ( V  \  {  .0.  } ) r  =  ( `' I `  ( N `  { z } ) ) ) )
57 breq1 4442 . . . . . 6  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( r ( le `  K ) ( `' I `  S )  <->  ( `' I `  ( N `  { z } ) ) ( le `  K ) ( `' I `  S ) ) )
58 oveq1 6277 . . . . . . 7  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( r (
join `  K )
( `' I `  ( N `  { Y } ) ) )  =  ( ( `' I `  ( N `
 { z } ) ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
5958breq2d 4451 . . . . . 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 708 . . . . 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 464 . . . 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 4654 . . 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 37474 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  V
)  ->  ( N `  { z } )  e.  ran  I )
6449, 51, 63syl2anc 659 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { z } )  e.  ran  I )
653adantr 463 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  S  e.  ran  I )
6626, 5, 6, 49, 64, 65dihcnvord 37417 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( `' I `  ( N `  { z } ) ) ( le `  K ) ( `' I `  S )  <->  ( N `  { z } ) 
C_  S ) )
6738adantr 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { Y } )  e.  ran  I )
6827, 5, 6, 41, 49, 64, 67djhj 37547 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( ( N `  { z } )  .\/  ( N `  { Y } ) ) )  =  ( ( `' I `  ( N `
 { z } ) ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
6968breq2d 4451 . . . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  X  e.  V )
7149, 70, 33syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { X } )  e.  ran  I )
725, 14, 6, 15dihrnss 37421 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { z } )  e.  ran  I )  ->  ( N `  { z } ) 
C_  V )
7349, 64, 72syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { z } )  C_  V
)
7440adantr 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { Y } )  C_  V
)
755, 6, 14, 15, 41djhcl 37543 . . . . . . . 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 1224 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( N `  {
z } )  .\/  ( N `  { Y } ) )  e. 
ran  I )
7726, 5, 6, 49, 71, 76dihcnvord 37417 . . . . . 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 255 . . . . 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 708 . . . 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 2962 . . 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 254 . 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 268 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    \ cdif 3458    C_ wss 3461   {csn 4016   class class class wbr 4439   `'ccnv 4987   ran crn 4989   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   0gc0g 14932   joincjn 15775   0.cp0 15869   LSpanclspn 17815   Atomscatm 35404   HLchlt 35491   LHypclh 36124   DVecHcdvh 37221   DIsoHcdih 37371  joinHcdjh 37537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 35100
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-undef 6994  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-sca 14803  df-vsca 14804  df-0g 14934  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-subg 16400  df-cntz 16557  df-lsm 16858  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-dvr 17530  df-drng 17596  df-lmod 17712  df-lss 17777  df-lsp 17816  df-lvec 17947  df-lsatoms 35117  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300  df-tendo 36897  df-edring 36899  df-disoa 37172  df-dvech 37222  df-dib 37282  df-dic 37316  df-dih 37372  df-doch 37491  df-djh 37538
This theorem is referenced by:  dihjat1lem  37571
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