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Theorem trlset 30972
Description: The set of traces of lattice translations for a fiducial co-atom  W. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
trlset.b  |-  B  =  ( Base `  K
)
trlset.l  |-  .<_  =  ( le `  K )
trlset.j  |-  .\/  =  ( join `  K )
trlset.m  |-  ./\  =  ( meet `  K )
trlset.a  |-  A  =  ( Atoms `  K )
trlset.h  |-  H  =  ( LHyp `  K
)
trlset.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlset.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlset  |-  ( ( K  e.  C  /\  W  e.  H )  ->  R  =  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) ) )
Distinct variable groups:    A, p    x, B    f, p, x, K    T, f    f, W, p, x
Allowed substitution hints:    A( x, f)    B( f, p)    C( x, f, p)    R( x, f, p)    T( x, p)    H( x, f, p)    .\/ ( x, f, p)    .<_ ( x, f, p)    ./\ (
x, f, p)

Proof of Theorem trlset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 trlset.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
2 trlset.b . . . . 5  |-  B  =  ( Base `  K
)
3 trlset.l . . . . 5  |-  .<_  =  ( le `  K )
4 trlset.j . . . . 5  |-  .\/  =  ( join `  K )
5 trlset.m . . . . 5  |-  ./\  =  ( meet `  K )
6 trlset.a . . . . 5  |-  A  =  ( Atoms `  K )
7 trlset.h . . . . 5  |-  H  =  ( LHyp `  K
)
82, 3, 4, 5, 6, 7trlfset 30971 . . . 4  |-  ( K  e.  C  ->  ( trL `  K )  =  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) ) )
98fveq1d 5543 . . 3  |-  ( K  e.  C  ->  (
( trL `  K
) `  W )  =  ( ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) ) `  W ) )
101, 9syl5eq 2340 . 2  |-  ( K  e.  C  ->  R  =  ( ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) ) `  W ) )
11 fveq2 5541 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
12 breq2 4043 . . . . . . . . 9  |-  ( w  =  W  ->  (
p  .<_  w  <->  p  .<_  W ) )
1312notbid 285 . . . . . . . 8  |-  ( w  =  W  ->  ( -.  p  .<_  w  <->  -.  p  .<_  W ) )
14 oveq2 5882 . . . . . . . . 9  |-  ( w  =  W  ->  (
( p  .\/  (
f `  p )
)  ./\  w )  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) )
1514eqeq2d 2307 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( ( p  .\/  ( f `
 p ) ) 
./\  w )  <->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) )
1613, 15imbi12d 311 . . . . . . 7  |-  ( w  =  W  ->  (
( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
)  <->  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )
1716ralbidv 2576 . . . . . 6  |-  ( w  =  W  ->  ( A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
)  <->  A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) )
1817riotabidv 6322 . . . . 5  |-  ( w  =  W  ->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( (
p  .\/  ( f `  p ) )  ./\  w ) ) )  =  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )
1911, 18mpteq12dv 4114 . . . 4  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  w ) ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) ) )
20 eqid 2296 . . . 4  |-  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) )  =  ( w  e.  H  |->  ( f  e.  ( ( LTrn `  K ) `  w
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  w
) ) ) ) )
21 fvex 5555 . . . . 5  |-  ( (
LTrn `  K ) `  W )  e.  _V
2221mptex 5762 . . . 4  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  ( f `  p
) )  ./\  W
) ) ) )  e.  _V
2319, 20, 22fvmpt 5618 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) ) `
 W )  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) ) )
24 trlset.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
25 eqid 2296 . . . 4  |-  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) )  =  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) )
2624, 25mpteq12i 4120 . . 3  |-  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) )  =  ( f  e.  ( ( LTrn `  K
) `  W )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) )
2723, 26syl6eqr 2346 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( f  e.  ( ( LTrn `  K
) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  w )
) ) ) ) `
 W )  =  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p  .\/  (
f `  p )
)  ./\  W )
) ) ) )
2810, 27sylan9eq 2348 1  |-  ( ( K  e.  C  /\  W  e.  H )  ->  R  =  ( f  e.  T  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  W  ->  x  =  ( ( p 
.\/  ( f `  p ) )  ./\  W ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  trlval  30973
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-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-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  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-ral 2561  df-rex 2562  df-reu 2563  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-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-riota 6320  df-trl 30970
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