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Theorem dih1 35294
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 35287 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  .1.  ) )
4 relxp 5058 . . 3  |-  Rel  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
)
5 eqid 2454 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
6 eqid 2454 . . . . 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 35095 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  V  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
109releqd 5035 . . 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 33370 . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  ( le
`  K )  =  ( le `  K
)
19 eqid 2454 . . . . . . . . . . . . . 14  |-  ( oc
`  K )  =  ( oc `  K
)
20 eqid 2454 . . . . . . . . . . . . . 14  |-  ( Atoms `  K )  =  (
Atoms `  K )
2118, 19, 20, 1lhpocnel 34025 . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . 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 34586 . . . . . . . . . . . 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 1219 . . . . . . . . . . 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 34774 . . . . . . . . . . 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 1219 . . . . . . . . . 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 34153 . . . . . . . . . 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 34726 . . . . . . . . 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 1219 . . . . . . . 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 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
33 eqid 2454 . . . . . . . . 9  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
3432, 1, 5, 33trlcl 34171 . . . . . . . 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 33199 . . . . . . 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 2524 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  V  <->  <. f ,  s >.  e.  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )
42 opelxp 4980 . . . . 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 33193 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
4644, 45syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .1.  e.  ( Base `  K ) )
47 hlpos 33373 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
4847adantr 465 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Poset )
4932, 1lhpbase 34005 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
5049adantl 466 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
51 eqid 2454 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5236, 51, 1lhp1cvr 34006 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
5332, 18, 51cvrnle 33288 . . . . . 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 1222 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  .1.  ( le
`  K ) W )
55 hlol 33369 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
56 eqid 2454 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
5732, 56, 36olm12 33236 . . . . . . . 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 6219 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  ( ( ( oc `  K ) `
 W ) (
join `  K ) W ) )
60 hllat 33371 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
6160adantr 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Lat )
6232, 19opoccl 33202 . . . . . . . 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 2454 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
6532, 64latjcom 15352 . . . . . . 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 1219 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) W )  =  ( W (
join `  K )
( ( oc `  K ) `  W
) ) )
6732, 19, 64, 36opexmid 33215 . . . . . . 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 2499 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  .1.  )
70 eqid 2454 . . . . . 6  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
71 vex 3081 . . . . . 6  |-  f  e. 
_V
72 vex 3081 . . . . . 6  |-  s  e. 
_V
7332, 18, 64, 56, 20, 1, 70, 5, 33, 6, 2, 23, 71, 72dihopelvalc 35257 . . . . 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 1228 . . . 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 5050 . 2  |-  ( ( ( Rel  ( I `
 .1.  )  /\  Rel  V )  /\  ( K  e.  HL  /\  W  e.  H ) )  -> 
( I `  .1.  )  =  V )
773, 11, 12, 76syl21anc 1218 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 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403    X. cxp 4949   `'ccnv 4950    o. ccom 4955   Rel wrel 4956   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14296   lecple 14368   occoc 14369   Posetcpo 15233   joincjn 15237   meetcmee 15238   1.cp1 15331   Latclat 15338   OPcops 33180   OLcol 33182    <o ccvr 33270   Atomscatm 33271   HLchlt 33358   LHypclh 33991   LTrncltrn 34108   trLctrl 34165   TEndoctendo 34759   DVecHcdvh 35086   DIsoHcdih 35236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-riotaBAD 32967
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-undef 6905  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-0g 14503  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-mnd 15538  df-submnd 15588  df-grp 15668  df-minusg 15669  df-sbg 15670  df-subg 15801  df-cntz 15958  df-lsm 16260  df-cmn 16404  df-abl 16405  df-mgp 16724  df-ur 16736  df-rng 16780  df-oppr 16848  df-dvdsr 16866  df-unit 16867  df-invr 16897  df-dvr 16908  df-drng 16967  df-lmod 17083  df-lss 17147  df-lsp 17186  df-lvec 17317  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166  df-tendo 34762  df-edring 34764  df-disoa 35037  df-dvech 35087  df-dib 35147  df-dic 35181  df-dih 35237
This theorem is referenced by:  dih1rn  35295  dih1cnv  35296  dihglb2  35350  doch0  35366  dochocss  35374
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