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Theorem dicffval 35971
Description: The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.)
Hypotheses
Ref Expression
dicval.l  |-  .<_  =  ( le `  K )
dicval.a  |-  A  =  ( Atoms `  K )
dicval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dicffval  |-  ( K  e.  V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e.  {
r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
Distinct variable groups:    A, r    w, H    f, g, q, r, s, w, K
Allowed substitution hints:    A( w, f, g, s, q)    H( f, g, s, r, q)    .<_ ( w, f, g, s, r, q)    V( w, f, g, s, r, q)

Proof of Theorem dicffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 3122 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5864 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2526 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5864 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
6 dicval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6syl6eqr 2526 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
8 fveq2 5864 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dicval.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2526 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4458 . . . . . . 7  |-  ( k  =  K  ->  (
r ( le `  k ) w  <->  r  .<_  w ) )
1211notbid 294 . . . . . 6  |-  ( k  =  K  ->  ( -.  r ( le `  k ) w  <->  -.  r  .<_  w ) )
137, 12rabeqbidv 3108 . . . . 5  |-  ( k  =  K  ->  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  =  { r  e.  A  |  -.  r  .<_  w }
)
14 fveq2 5864 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1514fveq1d 5866 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
16 fveq2 5864 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
1716fveq1d 5866 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  ( ( oc
`  K ) `  w ) )
1817fveq2d 5868 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
g `  ( ( oc `  k ) `  w ) )  =  ( g `  (
( oc `  K
) `  w )
) )
1918eqeq1d 2469 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( g `  (
( oc `  k
) `  w )
)  =  q  <->  ( g `  ( ( oc `  K ) `  w
) )  =  q ) )
2015, 19riotaeqbidv 6246 . . . . . . . . 9  |-  ( k  =  K  ->  ( iota_ g  e.  ( (
LTrn `  k ) `  w ) ( g `
 ( ( oc
`  k ) `  w ) )  =  q )  =  (
iota_ g  e.  (
( LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q ) )
2120fveq2d 5868 . . . . . . . 8  |-  ( k  =  K  ->  (
s `  ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) ) )
2221eqeq2d 2481 . . . . . . 7  |-  ( k  =  K  ->  (
f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  <->  f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K ) `  w
) ( g `  ( ( oc `  K ) `  w
) )  =  q ) ) ) )
23 fveq2 5864 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
2423fveq1d 5866 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
2524eleq2d 2537 . . . . . . 7  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w )  <->  s  e.  ( ( TEndo `  K
) `  w )
) )
2622, 25anbi12d 710 . . . . . 6  |-  ( k  =  K  ->  (
( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  k ) `  w
) ( g `  ( ( oc `  k ) `  w
) )  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
)  <->  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  w ) ) ) )
2726opabbidv 4510 . . . . 5  |-  ( k  =  K  ->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
) }  =  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  w ) ) } )
2813, 27mpteq12dv 4525 . . . 4  |-  ( k  =  K  ->  (
q  e.  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  k ) `  w ) ( g `
 ( ( oc
`  k ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  k ) `  w ) ) } )  =  ( q  e.  { r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) )
294, 28mpteq12dv 4525 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( q  e.  { r  e.  ( Atoms `  k )  |  -.  r ( le
`  k ) w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
) } ) )  =  ( w  e.  H  |->  ( q  e. 
{ r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
30 df-dic 35970 . . 3  |-  DIsoC  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( q  e.  { r  e.  ( Atoms `  k )  |  -.  r ( le
`  k ) w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
) } ) ) )
31 fvex 5874 . . . . 5  |-  ( LHyp `  K )  e.  _V
323, 31eqeltri 2551 . . . 4  |-  H  e. 
_V
3332mptex 6129 . . 3  |-  ( w  e.  H  |->  ( q  e.  { r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) )  e.  _V
3429, 30, 33fvmpt 5948 . 2  |-  ( K  e.  _V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e.  {
r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
351, 34syl 16 1  |-  ( K  e.  V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e.  {
r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   class class class wbr 4447   {copab 4504    |-> cmpt 4505   ` cfv 5586   iota_crio 6242   lecple 14555   occoc 14556   Atomscatm 34060   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   DIsoCcdic 35969
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-dic 35970
This theorem is referenced by:  dicfval  35972
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