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Theorem dih1 36484
Description: The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
Hypotheses
Ref Expression
dih1.m  |-  .1.  =  ( 1. `  K )
dih1.h  |-  H  =  ( LHyp `  K
)
dih1.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dih1.u  |-  U  =  ( ( DVecH `  K
) `  W )
dih1.v  |-  V  =  ( Base `  U
)
Assertion
Ref Expression
dih1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .1.  )  =  V )

Proof of Theorem dih1
Dummy variables  f 
g  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dih1.h . . 3  |-  H  =  ( LHyp `  K
)
2 dih1.i . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
31, 2dihvalrel 36477 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  .1.  ) )
4 relxp 5116 . . 3  |-  Rel  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
)
5 eqid 2467 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
6 eqid 2467 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dih1.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
8 dih1.v . . . . 5  |-  V  =  ( Base `  U
)
91, 5, 6, 7, 8dvhvbase 36285 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  V  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
109releqd 5093 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Rel  V  <->  Rel  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )
114, 10mpbiri 233 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  V )
12 id 22 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 hlop 34560 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1413ad2antrr 725 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  K  e.  OP )
15 simpl 457 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simprl 755 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  f  e.  ( ( LTrn `  K
) `  W )
)
17 simprr 756 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  s  e.  ( ( TEndo `  K ) `  W ) )
18 eqid 2467 . . . . . . . . . . . . . 14  |-  ( le
`  K )  =  ( le `  K
)
19 eqid 2467 . . . . . . . . . . . . . 14  |-  ( oc
`  K )  =  ( oc `  K
)
20 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Atoms `  K )  =  (
Atoms `  K )
2118, 19, 20, 1lhpocnel 35215 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )
2221adantr 465 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( ( ( oc `  K ) `
 W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
23 eqid 2467 . . . . . . . . . . . . 13  |-  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  ( ( oc `  K
) `  W )
)  =  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  ( ( oc `  K
) `  W )
)
2418, 20, 1, 5, 23ltrniotacl 35776 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( oc `  K ) `
 W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W )  /\  ( ( ( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )  ->  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) )  e.  ( ( LTrn `  K
) `  W )
)
2515, 22, 22, 24syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) )  e.  ( ( LTrn `  K ) `  W
) )
261, 5, 6tendocl 35964 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( ( TEndo `  K ) `  W )  /\  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) )  e.  ( (
LTrn `  K ) `  W ) )
2715, 17, 25, 26syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) )  e.  ( (
LTrn `  K ) `  W ) )
281, 5ltrncnv 35343 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) )  e.  ( (
LTrn `  K ) `  W ) )  ->  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) )  e.  ( (
LTrn `  K ) `  W ) )
2927, 28syldan 470 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  `' ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) )  e.  ( (
LTrn `  K ) `  W ) )
301, 5ltrnco 35916 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  ( ( LTrn `  K
) `  W )  /\  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) )  e.  ( ( LTrn `  K ) `  W
) )
3115, 16, 29, 30syl3anc 1228 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( f  o.  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) ) )  e.  ( ( LTrn `  K
) `  W )
)
32 eqid 2467 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
33 eqid 2467 . . . . . . . . 9  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
3432, 1, 5, 33trlcl 35361 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  o.  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) ) )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
( trL `  K
) `  W ) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) ) )  e.  ( Base `  K ) )
3531, 34syldan 470 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( ( ( trL `  K ) `
 W ) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) ) )  e.  ( Base `  K ) )
36 dih1.m . . . . . . . 8  |-  .1.  =  ( 1. `  K )
3732, 18, 36ople1 34389 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( ( ( trL `  K ) `  W
) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) ) ) )  e.  ( Base `  K
) )  ->  (
( ( trL `  K
) `  W ) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) ) ) ( le `  K )  .1.  )
3814, 35, 37syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( ( ( trL `  K ) `
 W ) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) ) ) ( le `  K )  .1.  )
3938ex 434 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) ) ) ) ( le `  K )  .1.  ) )
4039pm4.71d 634 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  <->  ( (
f  e.  ( (
LTrn `  K ) `  W )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  /\  ( (
( trL `  K
) `  W ) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) ) ) ( le `  K )  .1.  )
) )
419eleq2d 2537 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  V  <->  <. f ,  s >.  e.  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )
42 opelxp 5035 . . . . 5  |-  ( <.
f ,  s >.  e.  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  <->  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )
4341, 42syl6bb 261 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  V  <->  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) ) )
4413adantr 465 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  OP )
4532, 36op1cl 34383 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
4644, 45syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .1.  e.  ( Base `  K ) )
47 hlpos 34563 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
4847adantr 465 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Poset )
4932, 1lhpbase 35195 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
5049adantl 466 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
51 eqid 2467 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5236, 51, 1lhp1cvr 35196 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
5332, 18, 51cvrnle 34478 . . . . . 6  |-  ( ( ( K  e.  Poset  /\  W  e.  ( Base `  K )  /\  .1.  e.  ( Base `  K
) )  /\  W
(  <o  `  K )  .1.  )  ->  -.  .1.  ( le `  K ) W )
5448, 50, 46, 52, 53syl31anc 1231 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  .1.  ( le
`  K ) W )
55 hlol 34559 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
56 eqid 2467 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
5732, 56, 36olm12 34426 . . . . . . . 8  |-  ( ( K  e.  OL  /\  W  e.  ( Base `  K ) )  -> 
(  .1.  ( meet `  K ) W )  =  W )
5855, 49, 57syl2an 477 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .1.  ( meet `  K ) W )  =  W )
5958oveq2d 6311 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  ( ( ( oc `  K ) `
 W ) (
join `  K ) W ) )
60 hllat 34561 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
6160adantr 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Lat )
6232, 19opoccl 34392 . . . . . . . 8  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
6313, 49, 62syl2an 477 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
64 eqid 2467 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
6532, 64latjcom 15563 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  (
( ( oc `  K ) `  W
) ( join `  K
) W )  =  ( W ( join `  K ) ( ( oc `  K ) `
 W ) ) )
6661, 63, 50, 65syl3anc 1228 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) W )  =  ( W (
join `  K )
( ( oc `  K ) `  W
) ) )
6732, 19, 64, 36opexmid 34405 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( W ( join `  K ) ( ( oc `  K ) `
 W ) )  =  .1.  )
6813, 49, 67syl2an 477 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( W ( join `  K ) ( ( oc `  K ) `
 W ) )  =  .1.  )
6959, 66, 683eqtrd 2512 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  .1.  )
70 eqid 2467 . . . . . 6  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
71 vex 3121 . . . . . 6  |-  f  e. 
_V
72 vex 3121 . . . . . 6  |-  s  e. 
_V
7332, 18, 64, 56, 20, 1, 70, 5, 33, 6, 2, 23, 71, 72dihopelvalc 36447 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  .1.  e.  ( Base `  K )  /\  -.  .1.  ( le
`  K ) W )  /\  ( ( ( ( oc `  K ) `  W
)  e.  ( Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W )  /\  ( ( ( oc `  K ) `
 W ) (
join `  K )
(  .1.  ( meet `  K ) W ) )  =  .1.  )
)  ->  ( <. f ,  s >.  e.  ( I `  .1.  )  <->  ( ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) )  /\  ( ( ( trL `  K ) `  W
) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  ( ( oc `  K ) `
 W ) ) ) ) ) ( le `  K )  .1.  ) ) )
7412, 46, 54, 21, 69, 73syl122anc 1237 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  ( I `  .1.  )  <->  ( ( f  e.  ( ( LTrn `  K ) `  W
)  /\  s  e.  ( ( TEndo `  K
) `  W )
)  /\  ( (
( trL `  K
) `  W ) `  ( f  o.  `' ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) ) ) ) ) ( le `  K )  .1.  )
) )
7540, 43, 743bitr4rd 286 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  ( I `  .1.  )  <->  <. f ,  s
>.  e.  V ) )
7675eqrelrdv2 5108 . 2  |-  ( ( ( Rel  ( I `
 .1.  )  /\  Rel  V )  /\  ( K  e.  HL  /\  W  e.  H ) )  -> 
( I `  .1.  )  =  V )
773, 11, 12, 76syl21anc 1227 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .1.  )  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453    X. cxp 5003   `'ccnv 5004    o. ccom 5009   Rel wrel 5010   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Basecbs 14507   lecple 14579   occoc 14580   Posetcpo 15444   joincjn 15448   meetcmee 15449   1.cp1 15542   Latclat 15549   OPcops 34370   OLcol 34372    <o ccvr 34460   Atomscatm 34461   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   trLctrl 35355   TEndoctendo 35949   DVecHcdvh 36276   DIsoHcdih 36426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tendo 35952  df-edring 35954  df-disoa 36227  df-dvech 36277  df-dib 36337  df-dic 36371  df-dih 36427
This theorem is referenced by:  dih1rn  36485  dih1cnv  36486  dihglb2  36540  doch0  36556  dochocss  36564
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