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Theorem dicffval 31973
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 2965 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5729 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2487 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5729 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
6 dicval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6syl6eqr 2487 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
8 fveq2 5729 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dicval.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2487 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4224 . . . . . . 7  |-  ( k  =  K  ->  (
r ( le `  k ) w  <->  r  .<_  w ) )
1211notbid 287 . . . . . 6  |-  ( k  =  K  ->  ( -.  r ( le `  k ) w  <->  -.  r  .<_  w ) )
137, 12rabeqbidv 2952 . . . . 5  |-  ( k  =  K  ->  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  =  { r  e.  A  |  -.  r  .<_  w }
)
14 fveq2 5729 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1514fveq1d 5731 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
16 fveq2 5729 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
1716fveq1d 5731 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  ( ( oc
`  K ) `  w ) )
1817fveq2d 5733 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
g `  ( ( oc `  k ) `  w ) )  =  ( g `  (
( oc `  K
) `  w )
) )
1918eqeq1d 2445 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( g `  (
( oc `  k
) `  w )
)  =  q  <->  ( g `  ( ( oc `  K ) `  w
) )  =  q ) )
2015, 19riotaeqbidv 6553 . . . . . . . . 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 5733 . . . . . . . 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 2448 . . . . . . 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 5729 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
2423fveq1d 5731 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
2524eleq2d 2504 . . . . . . 7  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w )  <->  s  e.  ( ( TEndo `  K
) `  w )
) )
2622, 25anbi12d 693 . . . . . 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 4272 . . . . 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 4288 . . . 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 4288 . . 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 31972 . . 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 5743 . . . . 5  |-  ( LHyp `  K )  e.  _V
323, 31eqeltri 2507 . . . 4  |-  H  e. 
_V
3332mptex 5967 . . 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 5807 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957   class class class wbr 4213   {copab 4266    e. cmpt 4267   ` cfv 5455   iota_crio 6543   lecple 13537   occoc 13538   Atomscatm 30062   LHypclh 30782   LTrncltrn 30899   TEndoctendo 31550   DIsoCcdic 31971
This theorem is referenced by:  dicfval  31974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-riota 6550  df-dic 31972
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