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Theorem djhcvat42 35156
Description: A covering property. (cvrat42 33184 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 459 . . 3  |-  ( ph  ->  K  e.  HL )
3 djhcvat42.s . . . 4  |-  ( ph  ->  S  e.  ran  I
)
4 eqid 2443 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
5 djhcvat42.h . . . . 5  |-  H  =  ( LHyp `  K
)
6 djhcvat42.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
74, 5, 6dihcnvcl 35012 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  ran  I )  ->  ( `' I `  S )  e.  ( Base `  K
) )
81, 3, 7syl2anc 661 . . 3  |-  ( ph  ->  ( `' I `  S )  e.  (
Base `  K )
)
9 djhcvat42.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
109eldifad 3361 . . . 4  |-  ( ph  ->  X  e.  V )
11 eldifsni 4022 . . . . 5  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
129, 11syl 16 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
13 eqid 2443 . . . . 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 35074 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )
)
191, 10, 12, 18syl3anc 1218 . . 3  |-  ( ph  ->  ( `' I `  ( N `  { X } ) )  e.  ( Atoms `  K )
)
20 djhcvat42.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3361 . . . 4  |-  ( ph  ->  Y  e.  V )
22 eldifsni 4022 . . . . 5  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2320, 22syl 16 . . . 4  |-  ( ph  ->  Y  =/=  .0.  )
2413, 5, 14, 15, 16, 17, 6dihlspsnat 35074 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V  /\  Y  =/=  .0.  )  ->  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
)
251, 21, 23, 24syl3anc 1218 . . 3  |-  ( ph  ->  ( `' I `  ( N `  { Y } ) )  e.  ( Atoms `  K )
)
26 eqid 2443 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
27 eqid 2443 . . . 4  |-  ( join `  K )  =  (
join `  K )
28 eqid 2443 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
294, 26, 27, 28, 13cvrat42 33184 . . 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 1220 . 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 35026 . . . 4  |-  ( ph  ->  ( S  =  {  .0.  }  <->  ( `' I `  S )  =  ( 0. `  K ) ) )
3231necon3bid 2637 . . 3  |-  ( ph  ->  ( S  =/=  {  .0.  }  <->  ( `' I `  S )  =/=  ( 0. `  K ) ) )
335, 14, 15, 17, 6dihlsprn 35072 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( N `  { X } )  e.  ran  I )
341, 10, 33syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  { X } )  e.  ran  I )
355, 14, 6, 15dihrnss 35019 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  ran  I )  ->  S  C_  V )
361, 3, 35syl2anc 661 . . . . . 6  |-  ( ph  ->  S  C_  V )
375, 14, 15, 17, 6dihlsprn 35072 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( N `  { Y } )  e.  ran  I )
381, 21, 37syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  e.  ran  I )
395, 14, 6, 15dihrnss 35019 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { Y } )  e. 
ran  I )  -> 
( N `  { Y } )  C_  V
)
401, 38, 39syl2anc 661 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  C_  V
)
41 djhcvat42.j . . . . . . 7  |-  .\/  =  ( (joinH `  K ) `  W )
425, 6, 14, 15, 41djhcl 35141 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  V  /\  ( N `  { Y } )  C_  V ) )  -> 
( S  .\/  ( N `  { Y } ) )  e. 
ran  I )
431, 36, 40, 42syl12anc 1216 . . . . 5  |-  ( ph  ->  ( S  .\/  ( N `  { Y } ) )  e. 
ran  I )
4426, 5, 6, 1, 34, 43dihcnvord 35015 . . . 4  |-  ( ph  ->  ( ( `' I `  ( N `  { X } ) ) ( le `  K ) ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  <->  ( N `  { X } ) 
C_  ( S  .\/  ( N `  { Y } ) ) ) )
4527, 5, 6, 41, 1, 3, 38djhj 35145 . . . . 5  |-  ( ph  ->  ( `' I `  ( S  .\/  ( N `
 { Y }
) ) )  =  ( ( `' I `  S ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
4645breq2d 4325 . . . 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 710 . 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 465 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
50 eldifi 3499 . . . . . 6  |-  ( z  e.  ( V  \  {  .0.  } )  -> 
z  e.  V )
5150adantl 466 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  z  e.  V )
52 eldifsni 4022 . . . . . 6  |-  ( z  e.  ( V  \  {  .0.  } )  -> 
z  =/=  .0.  )
5352adantl 466 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  z  =/=  .0.  )
5413, 5, 14, 15, 16, 17, 6dihlspsnat 35074 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  V  /\  z  =/=  .0.  )  ->  ( `' I `  ( N `  {
z } ) )  e.  ( Atoms `  K
) )
5549, 51, 53, 54syl3anc 1218 . . . 4  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( N `
 { z } ) )  e.  (
Atoms `  K ) )
5613, 5, 14, 15, 16, 17, 6, 1dihatexv2 35080 . . . 4  |-  ( ph  ->  ( r  e.  (
Atoms `  K )  <->  E. z  e.  ( V  \  {  .0.  } ) r  =  ( `' I `  ( N `  { z } ) ) ) )
57 breq1 4316 . . . . . 6  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( r ( le `  K ) ( `' I `  S )  <->  ( `' I `  ( N `  { z } ) ) ( le `  K ) ( `' I `  S ) ) )
58 oveq1 6119 . . . . . . 7  |-  ( r  =  ( `' I `  ( N `  {
z } ) )  ->  ( r (
join `  K )
( `' I `  ( N `  { Y } ) ) )  =  ( ( `' I `  ( N `
 { z } ) ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
5958breq2d 4325 . . . . . 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 710 . . . . 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 466 . . . 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 4530 . . 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 35072 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  V
)  ->  ( N `  { z } )  e.  ran  I )
6449, 51, 63syl2anc 661 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { z } )  e.  ran  I )
653adantr 465 . . . . . 6  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  S  e.  ran  I )
6626, 5, 6, 49, 64, 65dihcnvord 35015 . . . . 5  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( `' I `  ( N `  { z } ) ) ( le `  K ) ( `' I `  S )  <->  ( N `  { z } ) 
C_  S ) )
6738adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { Y } )  e.  ran  I )
6827, 5, 6, 41, 49, 64, 67djhj 35145 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( `' I `  ( ( N `  { z } )  .\/  ( N `  { Y } ) ) )  =  ( ( `' I `  ( N `
 { z } ) ) ( join `  K ) ( `' I `  ( N `
 { Y }
) ) ) )
6968breq2d 4325 . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  X  e.  V )
7149, 70, 33syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { X } )  e.  ran  I )
725, 14, 6, 15dihrnss 35019 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { z } )  e.  ran  I )  ->  ( N `  { z } ) 
C_  V )
7349, 64, 72syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { z } )  C_  V
)
7440adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  ( N `  { Y } )  C_  V
)
755, 6, 14, 15, 41djhcl 35141 . . . . . . . 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 1216 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( V  \  {  .0.  } ) )  ->  (
( N `  {
z } )  .\/  ( N `  { Y } ) )  e. 
ran  I )
7726, 5, 6, 49, 71, 76dihcnvord 35015 . . . . . 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 710 . . . 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 2753 . . 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 369    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2737    \ cdif 3346    C_ wss 3349   {csn 3898   class class class wbr 4313   `'ccnv 4860   ran crn 4862   ` cfv 5439  (class class class)co 6112   Basecbs 14195   lecple 14266   0gc0g 14399   joincjn 15135   0.cp0 15228   LSpanclspn 17074   Atomscatm 33004   HLchlt 33091   LHypclh 33724   DVecHcdvh 34819   DIsoHcdih 34969  joinHcdjh 35135
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-undef 6813  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-0g 14401  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-drng 16856  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lvec 17206  df-lsatoms 32717  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-llines 33238  df-lplanes 33239  df-lvols 33240  df-lines 33241  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899  df-tendo 34495  df-edring 34497  df-disoa 34770  df-dvech 34820  df-dib 34880  df-dic 34914  df-dih 34970  df-doch 35089  df-djh 35136
This theorem is referenced by:  dihjat1lem  35169
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