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Theorem dicffval 31364
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 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2333 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
6 dicval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
8 fveq2 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dicval.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2333 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4034 . . . . . . 7  |-  ( k  =  K  ->  (
r ( le `  k ) w  <->  r  .<_  w ) )
1211notbid 285 . . . . . 6  |-  ( k  =  K  ->  ( -.  r ( le `  k ) w  <->  -.  r  .<_  w ) )
137, 12rabeqbidv 2783 . . . . 5  |-  ( k  =  K  ->  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  =  { r  e.  A  |  -.  r  .<_  w }
)
14 fveq2 5525 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1514fveq1d 5527 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
16 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
1716fveq1d 5527 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  ( ( oc
`  K ) `  w ) )
1817fveq2d 5529 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
g `  ( ( oc `  k ) `  w ) )  =  ( g `  (
( oc `  K
) `  w )
) )
1918eqeq1d 2291 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( g `  (
( oc `  k
) `  w )
)  =  q  <->  ( g `  ( ( oc `  K ) `  w
) )  =  q ) )
2015, 19riotaeqbidv 6307 . . . . . . . . 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 5529 . . . . . . . 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 2294 . . . . . . 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 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
2423fveq1d 5527 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
2524eleq2d 2350 . . . . . . 7  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w )  <->  s  e.  ( ( TEndo `  K
) `  w )
) )
2622, 25anbi12d 691 . . . . . 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 4082 . . . . 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 4098 . . . 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 4098 . . 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 31363 . . 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 5539 . . . . 5  |-  ( LHyp `  K )  e.  _V
323, 31eqeltri 2353 . . . 4  |-  H  e. 
_V
3332mptex 5746 . . 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 5602 . 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 15 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023   {copab 4076    e. cmpt 4077   ` cfv 5255   iota_crio 6297   lecple 13215   occoc 13216   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DIsoCcdic 31362
This theorem is referenced by:  dicfval  31365
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-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-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-riota 6304  df-dic 31363
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