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Theorem dicffval 34816
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 2979 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5689 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2491 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5689 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
6 dicval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6syl6eqr 2491 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
8 fveq2 5689 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dicval.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2491 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4301 . . . . . . 7  |-  ( k  =  K  ->  (
r ( le `  k ) w  <->  r  .<_  w ) )
1211notbid 294 . . . . . 6  |-  ( k  =  K  ->  ( -.  r ( le `  k ) w  <->  -.  r  .<_  w ) )
137, 12rabeqbidv 2965 . . . . 5  |-  ( k  =  K  ->  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  =  { r  e.  A  |  -.  r  .<_  w }
)
14 fveq2 5689 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1514fveq1d 5691 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
16 fveq2 5689 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
1716fveq1d 5691 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  ( ( oc
`  K ) `  w ) )
1817fveq2d 5693 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
g `  ( ( oc `  k ) `  w ) )  =  ( g `  (
( oc `  K
) `  w )
) )
1918eqeq1d 2449 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( g `  (
( oc `  k
) `  w )
)  =  q  <->  ( g `  ( ( oc `  K ) `  w
) )  =  q ) )
2015, 19riotaeqbidv 6053 . . . . . . . . 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 5693 . . . . . . . 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 2452 . . . . . . 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 5689 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
2423fveq1d 5691 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
2524eleq2d 2508 . . . . . . 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 4353 . . . . 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 4368 . . . 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 4368 . . 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 34815 . . 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 5699 . . . . 5  |-  ( LHyp `  K )  e.  _V
323, 31eqeltri 2511 . . . 4  |-  H  e. 
_V
3332mptex 5946 . . 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 5772 . 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 1369    e. wcel 1756   {crab 2717   _Vcvv 2970   class class class wbr 4290   {copab 4347    e. cmpt 4348   ` cfv 5416   iota_crio 6049   lecple 14243   occoc 14244   Atomscatm 32905   LHypclh 33625   LTrncltrn 33742   TEndoctendo 34393   DIsoCcdic 34814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-dic 34815
This theorem is referenced by:  dicfval  34817
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