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Theorem dihlspsnat 34333
Description: The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
Hypotheses
Ref Expression
dihlspsnat.a  |-  A  =  ( Atoms `  K )
dihlspsnat.h  |-  H  =  ( LHyp `  K
)
dihlspsnat.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihlspsnat.v  |-  V  =  ( Base `  U
)
dihlspsnat.o  |-  .0.  =  ( 0g `  U )
dihlspsnat.n  |-  N  =  ( LSpan `  U )
dihlspsnat.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihlspsnat  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  A )

Proof of Theorem dihlspsnat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2 dihlspsnat.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 dihlspsnat.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
4 dihlspsnat.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2402 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
61, 2, 3, 4, 5dihf11 34267 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : ( Base `  K ) -1-1-> ( LSubSp `  U ) )
763ad2ant1 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  I : (
Base `  K ) -1-1-> ( LSubSp `  U )
)
8 f1f1orn 5809 . . . 4  |-  ( I : ( Base `  K
) -1-1-> ( LSubSp `  U
)  ->  I :
( Base `  K ) -1-1-onto-> ran  I )
97, 8syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  I : (
Base `  K ) -1-1-onto-> ran  I )
10 dihlspsnat.v . . . . 5  |-  V  =  ( Base `  U
)
11 dihlspsnat.n . . . . 5  |-  N  =  ( LSpan `  U )
122, 4, 10, 11, 3dihlsprn 34331 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( N `  { X } )  e.  ran  I )
13123adant3 1017 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( N `  { X } )  e. 
ran  I )
14 f1ocnvdm 6170 . . 3  |-  ( ( I : ( Base `  K ) -1-1-onto-> ran  I  /\  ( N `  { X } )  e.  ran  I )  ->  ( `' I `  ( N `
 { X }
) )  e.  (
Base `  K )
)
159, 13, 14syl2anc 659 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  ( Base `  K
) )
16 fveq2 5848 . . . . 5  |-  ( ( `' I `  ( N `
 { X }
) )  =  ( 0. `  K )  ->  ( I `  ( `' I `  ( N `
 { X }
) ) )  =  ( I `  ( 0. `  K ) ) )
172, 3dihcnvid2 34273 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( N `  { X } )  e. 
ran  I )  -> 
( I `  ( `' I `  ( N `
 { X }
) ) )  =  ( N `  { X } ) )
1812, 17syldan 468 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( I `  ( `' I `  ( N `  { X } ) ) )  =  ( N `  { X } ) )
19 eqid 2402 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
20 dihlspsnat.o . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
2119, 2, 3, 4, 20dih0 34280 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  ( 0. `  K ) )  =  {  .0.  }
)
2221adantr 463 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( I `  ( 0. `  K
) )  =  {  .0.  } )
2318, 22eqeq12d 2424 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( (
I `  ( `' I `  ( N `  { X } ) ) )  =  ( I `  ( 0.
`  K ) )  <-> 
( N `  { X } )  =  {  .0.  } ) )
24 id 22 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
252, 4, 24dvhlmod 34110 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
2610, 20, 11lspsneq0 17976 . . . . . . 7  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  (
( N `  { X } )  =  {  .0.  }  <->  X  =  .0.  ) )
2725, 26sylan 469 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( ( N `  { X } )  =  {  .0.  }  <->  X  =  .0.  ) )
2823, 27bitrd 253 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( (
I `  ( `' I `  ( N `  { X } ) ) )  =  ( I `  ( 0.
`  K ) )  <-> 
X  =  .0.  )
)
2916, 28syl5ib 219 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( ( `' I `  ( N `
 { X }
) )  =  ( 0. `  K )  ->  X  =  .0.  ) )
3029necon3d 2627 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V
)  ->  ( X  =/=  .0.  ->  ( `' I `  ( N `  { X } ) )  =/=  ( 0.
`  K ) ) )
31303impia 1194 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  =/=  ( 0. `  K
) )
32 simpll1 1036 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
332, 4, 32dvhlvec 34109 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  ->  U  e.  LVec )
34 simplr 754 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  ->  x  e.  ( Base `  K ) )
351, 2, 3, 4, 5dihlss 34250 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  (
Base `  K )
)  ->  ( I `  x )  e.  (
LSubSp `  U ) )
3632, 34, 35syl2anc 659 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  -> 
( I `  x
)  e.  ( LSubSp `  U ) )
37 simpll2 1037 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  ->  X  e.  V )
38 simpr 459 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  -> 
( I `  x
)  C_  ( N `  { X } ) )
3910, 20, 5, 11lspsnat 18109 . . . . . 6  |-  ( ( ( U  e.  LVec  /\  ( I `  x
)  e.  ( LSubSp `  U )  /\  X  e.  V )  /\  (
I `  x )  C_  ( N `  { X } ) )  -> 
( ( I `  x )  =  ( N `  { X } )  \/  (
I `  x )  =  {  .0.  } ) )
4033, 36, 37, 38, 39syl31anc 1233 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  /\  (
I `  x )  C_  ( N `  { X } ) )  -> 
( ( I `  x )  =  ( N `  { X } )  \/  (
I `  x )  =  {  .0.  } ) )
4140ex 432 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  C_  ( N `  { X } )  ->  ( ( I `
 x )  =  ( N `  { X } )  \/  (
I `  x )  =  {  .0.  } ) ) )
42 simp1 997 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
4342, 13, 17syl2anc 659 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( I `  ( `' I `  ( N `
 { X }
) ) )  =  ( N `  { X } ) )
4443adantr 463 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
I `  ( `' I `  ( N `  { X } ) ) )  =  ( N `  { X } ) )
4544sseq2d 3469 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  C_  ( I `  ( `' I `  ( N `  { X } ) ) )  <-> 
( I `  x
)  C_  ( N `  { X } ) ) )
46 simpl1 1000 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
47 simpr 459 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  x  e.  ( Base `  K
) )
4815adantr 463 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  ( `' I `  ( N `
 { X }
) )  e.  (
Base `  K )
)
49 eqid 2402 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
501, 49, 2, 3dihord 34264 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  (
Base `  K )  /\  ( `' I `  ( N `  { X } ) )  e.  ( Base `  K
) )  ->  (
( I `  x
)  C_  ( I `  ( `' I `  ( N `  { X } ) ) )  <-> 
x ( le `  K ) ( `' I `  ( N `
 { X }
) ) ) )
5146, 47, 48, 50syl3anc 1230 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  C_  ( I `  ( `' I `  ( N `  { X } ) ) )  <-> 
x ( le `  K ) ( `' I `  ( N `
 { X }
) ) ) )
5245, 51bitr3d 255 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  C_  ( N `  { X } )  <-> 
x ( le `  K ) ( `' I `  ( N `
 { X }
) ) ) )
5344eqeq2d 2416 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  ( I `
 ( `' I `  ( N `  { X } ) ) )  <-> 
( I `  x
)  =  ( N `
 { X }
) ) )
541, 2, 3dih11 34265 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  (
Base `  K )  /\  ( `' I `  ( N `  { X } ) )  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  ( I `
 ( `' I `  ( N `  { X } ) ) )  <-> 
x  =  ( `' I `  ( N `
 { X }
) ) ) )
5546, 47, 48, 54syl3anc 1230 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  ( I `
 ( `' I `  ( N `  { X } ) ) )  <-> 
x  =  ( `' I `  ( N `
 { X }
) ) ) )
5653, 55bitr3d 255 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  ( N `
 { X }
)  <->  x  =  ( `' I `  ( N `
 { X }
) ) ) )
5746, 21syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
I `  ( 0. `  K ) )  =  {  .0.  } )
5857eqeq2d 2416 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  ( I `
 ( 0. `  K ) )  <->  ( I `  x )  =  {  .0.  } ) )
59 simpl1l 1048 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  K  e.  HL )
60 hlop 32360 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
611, 19op0cl 32182 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
6259, 60, 613syl 20 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  ( 0. `  K )  e.  ( Base `  K
) )
631, 2, 3dih11 34265 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  (
Base `  K )  /\  ( 0. `  K
)  e.  ( Base `  K ) )  -> 
( ( I `  x )  =  ( I `  ( 0.
`  K ) )  <-> 
x  =  ( 0.
`  K ) ) )
6446, 47, 62, 63syl3anc 1230 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  ( I `
 ( 0. `  K ) )  <->  x  =  ( 0. `  K ) ) )
6558, 64bitr3d 255 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( I `  x
)  =  {  .0.  }  <-> 
x  =  ( 0.
`  K ) ) )
6656, 65orbi12d 708 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
( ( I `  x )  =  ( N `  { X } )  \/  (
I `  x )  =  {  .0.  } )  <-> 
( x  =  ( `' I `  ( N `
 { X }
) )  \/  x  =  ( 0. `  K ) ) ) )
6741, 52, 663imtr3d 267 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/= 
.0.  )  /\  x  e.  ( Base `  K
) )  ->  (
x ( le `  K ) ( `' I `  ( N `
 { X }
) )  ->  (
x  =  ( `' I `  ( N `
 { X }
) )  \/  x  =  ( 0. `  K ) ) ) )
6867ralrimiva 2817 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  A. x  e.  (
Base `  K )
( x ( le
`  K ) ( `' I `  ( N `
 { X }
) )  ->  (
x  =  ( `' I `  ( N `
 { X }
) )  \/  x  =  ( 0. `  K ) ) ) )
69 simp1l 1021 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  K  e.  HL )
70 hlatl 32358 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
71 dihlspsnat.a . . . 4  |-  A  =  ( Atoms `  K )
721, 49, 19, 71isat3 32305 . . 3  |-  ( K  e.  AtLat  ->  ( ( `' I `  ( N `
 { X }
) )  e.  A  <->  ( ( `' I `  ( N `  { X } ) )  e.  ( Base `  K
)  /\  ( `' I `  ( N `  { X } ) )  =/=  ( 0.
`  K )  /\  A. x  e.  ( Base `  K ) ( x ( le `  K
) ( `' I `  ( N `  { X } ) )  -> 
( x  =  ( `' I `  ( N `
 { X }
) )  \/  x  =  ( 0. `  K ) ) ) ) ) )
7369, 70, 723syl 20 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( ( `' I `  ( N `
 { X }
) )  e.  A  <->  ( ( `' I `  ( N `  { X } ) )  e.  ( Base `  K
)  /\  ( `' I `  ( N `  { X } ) )  =/=  ( 0.
`  K )  /\  A. x  e.  ( Base `  K ) ( x ( le `  K
) ( `' I `  ( N `  { X } ) )  -> 
( x  =  ( `' I `  ( N `
 { X }
) )  \/  x  =  ( 0. `  K ) ) ) ) ) )
7415, 31, 68, 73mpbir3and 1180 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753    C_ wss 3413   {csn 3971   class class class wbr 4394   `'ccnv 4821   ran crn 4823   -1-1->wf1 5565   -1-1-onto->wf1o 5567   ` cfv 5568   Basecbs 14839   lecple 14914   0gc0g 15052   0.cp0 15989   LModclmod 17830   LSubSpclss 17896   LSpanclspn 17935   LVecclvec 18066   OPcops 32170   Atomscatm 32261   AtLatcal 32262   HLchlt 32348   LHypclh 32981   DVecHcdvh 34078   DIsoHcdih 34228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-tpos 6957  df-undef 7004  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-0g 15054  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-cntz 16677  df-lsm 16978  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-dvr 17650  df-drng 17716  df-lmod 17832  df-lss 17897  df-lsp 17936  df-lvec 18067  df-lsatoms 31974  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tendo 33754  df-edring 33756  df-disoa 34029  df-dvech 34079  df-dib 34139  df-dic 34173  df-dih 34229
This theorem is referenced by:  dihlatat  34337  djhcvat42  34415  dihprrnlem1N  34424  dihprrnlem2  34425
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