Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dih1 Unicode version

Theorem dih1 31476
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 31469 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  .1.  ) )
4 relxp 4794 . . 3  |-  Rel  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
)
5 eqid 2283 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
6 eqid 2283 . . . . 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 31277 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  V  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
109releqd 4773 . . 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 29552 . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( le
`  K )  =  ( le `  K
)
19 eqid 2283 . . . . . . . . . . . . . 14  |-  ( oc
`  K )  =  ( oc `  K
)
20 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Atoms `  K )  =  (
Atoms `  K )
2118, 19, 20, 1lhpocnel 30207 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 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 30768 . . . . . . . . . . . 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 30956 . . . . . . . . . . 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 30335 . . . . . . . . . 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 30908 . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
33 eqid 2283 . . . . . . . . 9  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
3432, 1, 5, 33trlcl 30353 . . . . . . . 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 29381 . . . . . . 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 2350 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. f ,  s
>.  e.  V  <->  <. f ,  s >.  e.  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )
42 opelxp 4719 . . . . 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 29375 . . . . . 6  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
4644, 45syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .1.  e.  ( Base `  K ) )
47 hlpos 29555 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
4847adantr 451 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Poset )
4932, 1lhpbase 30187 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
5049adantl 452 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
51 eqid 2283 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5236, 51, 1lhp1cvr 30188 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
5332, 18, 51cvrnle 29470 . . . . . 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 29551 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
56 eqid 2283 . . . . . . . . 9  |-  ( meet `  K )  =  (
meet `  K )
5732, 56, 36olm12 29418 . . . . . . . 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 5874 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  ( ( ( oc `  K ) `
 W ) (
join `  K ) W ) )
60 hllat 29553 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
6160adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  Lat )
6232, 19opoccl 29384 . . . . . . . 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 2283 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
6532, 64latjcom 14165 . . . . . . 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 29397 . . . . . . 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 2319 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W ) ( join `  K ) (  .1.  ( meet `  K
) W ) )  =  .1.  )
70 eqid 2283 . . . . . 6  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
71 vex 2791 . . . . . 6  |-  f  e. 
_V
72 vex 2791 . . . . . 6  |-  s  e. 
_V
7332, 18, 64, 56, 20, 1, 70, 5, 33, 6, 2, 23, 71, 72dihopelvalc 31439 . . . . 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 4786 . 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 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    X. cxp 4687   `'ccnv 4688    o. ccom 4693   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   occoc 13216   Posetcpo 14074   joincjn 14078   meetcmee 14079   1.cp1 14144   Latclat 14151   OPcops 29362   OLcol 29364    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941   DVecHcdvh 31268   DIsoHcdih 31418
This theorem is referenced by:  dih1rn  31477  dih1cnv  31478  dihglb2  31532  doch0  31548  dochocss  31556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419
  Copyright terms: Public domain W3C validator