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Theorem dih1 32098
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 32091 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  .1.  ) )
4 relxp 4810 . . 3  |-  Rel  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
)
5 eqid 2296 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
6 eqid 2296 . . . . 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 31899 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  V  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
109releqd 4789 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Rel  V  <->  Rel  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )
114, 10mpbiri 224 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  V )
12 id 19 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 hlop 30174 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1413ad2antrr 706 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  K  e.  OP )
15 simpl 443 . . . . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  f  e.  ( ( LTrn `  K
) `  W )
)
17 simprr 733 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )  ->  s  e.  ( ( TEndo `  K ) `  W ) )
18 eqid 2296 . . . . . . . . . . . . . 14  |-  ( le
`  K )  =  ( le `  K
)
19 eqid 2296 . . . . . . . . . . . . . 14  |-  ( oc
`  K )  =  ( oc `  K
)
20 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Atoms `  K )  =  (
Atoms `  K )
2118, 19, 20, 1lhpocnel 30829 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )
2221adantr 451 . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 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 31390 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 31578 . . . . . . . . . . 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 1182 . . . . . . . . . 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 30957 . . . . . . . . . 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 456 . . . . . . . . 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 31530 . . . . . . . . 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 1182 . . . . . . . 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 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
33 eqid 2296 . . . . . . . . 9  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
3432, 1, 5, 33trlcl 30975 . . . . . . . 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 456 . . . . . . 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 30003 . . . . . . 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 642 . . . . . 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 423 . . . . 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 615 . . . 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 2363 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  V  <->  <. f ,  s >.  e.  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )
42 opelxp 4735 . . . . 5  |-  ( <.
f ,  s >.  e.  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  <->  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) )
4341, 42syl6bb 252 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  V  <->  ( f  e.  ( ( LTrn `  K
) `  W )  /\  s  e.  (
( TEndo `  K ) `  W ) ) ) )
4413adantr 451 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  OP )
4532, 36op1cl 29997 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
4644, 45syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .1.  e.  ( Base `  K ) )
47 hlpos 30177 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
4847adantr 451 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Poset )
4932, 1lhpbase 30809 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
5049adantl 452 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
51 eqid 2296 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5236, 51, 1lhp1cvr 30810 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
5332, 18, 51cvrnle 30092 . . . . . 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 1185 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  .1.  ( le
`  K ) W )
55 hlol 30173 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
56 eqid 2296 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
5732, 56, 36olm12 30040 . . . . . . . 8  |-  ( ( K  e.  OL  /\  W  e.  ( Base `  K ) )  -> 
(  .1.  ( meet `  K ) W )  =  W )
5855, 49, 57syl2an 463 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .1.  ( meet `  K ) W )  =  W )
5958oveq2d 5890 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  ( ( ( oc `  K ) `
 W ) (
join `  K ) W ) )
60 hllat 30175 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
6160adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Lat )
6232, 19opoccl 30006 . . . . . . . 8  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
6313, 49, 62syl2an 463 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
64 eqid 2296 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
6532, 64latjcom 14181 . . . . . . 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 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) W )  =  ( W (
join `  K )
( ( oc `  K ) `  W
) ) )
6732, 19, 64, 36opexmid 30019 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( W ( join `  K ) ( ( oc `  K ) `
 W ) )  =  .1.  )
6813, 49, 67syl2an 463 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( W ( join `  K ) ( ( oc `  K ) `
 W ) )  =  .1.  )
6959, 66, 683eqtrd 2332 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  .1.  )
70 eqid 2296 . . . . . 6  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
71 vex 2804 . . . . . 6  |-  f  e. 
_V
72 vex 2804 . . . . . 6  |-  s  e. 
_V
7332, 18, 64, 56, 20, 1, 70, 5, 33, 6, 2, 23, 71, 72dihopelvalc 32061 . . . . 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 1191 . . . 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 277 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  ( I `  .1.  )  <->  <. f ,  s
>.  e.  V ) )
7675eqrelrdv2 4802 . 2  |-  ( ( ( Rel  ( I `
 .1.  )  /\  Rel  V )  /\  ( K  e.  HL  /\  W  e.  H ) )  -> 
( I `  .1.  )  =  V )
773, 11, 12, 76syl21anc 1181 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .1.  )  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    X. cxp 4703   `'ccnv 4704    o. ccom 4709   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   occoc 13232   Posetcpo 14090   joincjn 14094   meetcmee 14095   1.cp1 14160   Latclat 14167   OPcops 29984   OLcol 29986    <o ccvr 30074   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563   DVecHcdvh 31890   DIsoHcdih 32040
This theorem is referenced by:  dih1rn  32099  dih1cnv  32100  dihglb2  32154  doch0  32170  dochocss  32178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566  df-edring 31568  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041
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