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Theorem dicfval 34825
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
)
dicval.p  |-  P  =  ( ( oc `  K ) `  W
)
dicval.t  |-  T  =  ( ( LTrn `  K
) `  W )
dicval.e  |-  E  =  ( ( TEndo `  K
) `  W )
dicval.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicfval  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) } ) )
Distinct variable groups:    A, r    f, g, q, r, s, K    .<_ , q    A, q    T, g    f, W, g, q, r, s
Allowed substitution hints:    A( f, g, s)    P( f, g, s, r, q)    T( f, s, r, q)    E( f, g, s, r, q)    H( f, g, s, r, q)    I( f, g, s, r, q)    .<_ ( f, g, s, r)    V( f, g, s, r, q)

Proof of Theorem dicfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dicval.i . . 3  |-  I  =  ( ( DIsoC `  K
) `  W )
2 dicval.l . . . . 5  |-  .<_  =  ( le `  K )
3 dicval.a . . . . 5  |-  A  =  ( Atoms `  K )
4 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
52, 3, 4dicffval 34824 . . . 4  |-  ( 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 )
) } ) ) )
65fveq1d 5698 . . 3  |-  ( K  e.  V  ->  (
( DIsoC `  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 )
) } ) ) `
 W ) )
71, 6syl5eq 2487 . 2  |-  ( K  e.  V  ->  I  =  ( ( 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 )
) } ) ) `
 W ) )
8 breq2 4301 . . . . . 6  |-  ( w  =  W  ->  (
r  .<_  w  <->  r  .<_  W ) )
98notbid 294 . . . . 5  |-  ( w  =  W  ->  ( -.  r  .<_  w  <->  -.  r  .<_  W ) )
109rabbidv 2969 . . . 4  |-  ( w  =  W  ->  { r  e.  A  |  -.  r  .<_  w }  =  { r  e.  A  |  -.  r  .<_  W }
)
11 fveq2 5696 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
12 dicval.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
1311, 12syl6eqr 2493 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
14 fveq2 5696 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( oc `  K
) `  w )  =  ( ( oc
`  K ) `  W ) )
15 dicval.p . . . . . . . . . . . 12  |-  P  =  ( ( oc `  K ) `  W
)
1614, 15syl6eqr 2493 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( oc `  K
) `  w )  =  P )
1716fveq2d 5700 . . . . . . . . . 10  |-  ( w  =  W  ->  (
g `  ( ( oc `  K ) `  w ) )  =  ( g `  P
) )
1817eqeq1d 2451 . . . . . . . . 9  |-  ( w  =  W  ->  (
( g `  (
( oc `  K
) `  w )
)  =  q  <->  ( g `  P )  =  q ) )
1913, 18riotaeqbidv 6060 . . . . . . . 8  |-  ( w  =  W  ->  ( iota_ g  e.  ( (
LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q )  =  (
iota_ g  e.  T  ( g `  P
)  =  q ) )
2019fveq2d 5700 . . . . . . 7  |-  ( w  =  W  ->  (
s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) ) )
2120eqeq2d 2454 . . . . . 6  |-  ( w  =  W  ->  (
f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  <->  f  =  ( s `  ( iota_ g  e.  T  ( g `
 P )  =  q ) ) ) )
22 fveq2 5696 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
23 dicval.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
2422, 23syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
2524eleq2d 2510 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w )  <->  s  e.  E ) )
2621, 25anbi12d 710 . . . . 5  |-  ( w  =  W  ->  (
( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K ) `  w
) ( g `  ( ( oc `  K ) `  w
) )  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
)  <->  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E
) ) )
2726opabbidv 4360 . . . 4  |-  ( w  =  W  ->  { <. 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.  T  ( g `  P )  =  q ) )  /\  s  e.  E
) } )
2810, 27mpteq12dv 4375 . . 3  |-  ( w  =  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 ) ) } )  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) } ) )
29 eqid 2443 . . 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 )
) } ) )  =  ( 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 fvex 5706 . . . . . 6  |-  ( Atoms `  K )  e.  _V
313, 30eqeltri 2513 . . . . 5  |-  A  e. 
_V
3231rabex 4448 . . . 4  |-  { r  e.  A  |  -.  r  .<_  W }  e.  _V
3332mptex 5953 . . 3  |-  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) } )  e.  _V
3428, 29, 33fvmpt 5779 . 2  |-  ( W  e.  H  ->  (
( 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 )
) } ) ) `
 W )  =  ( q  e.  {
r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  q ) )  /\  s  e.  E ) } ) )
357, 34sylan9eq 2495 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977   class class class wbr 4297   {copab 4354    e. cmpt 4355   ` cfv 5423   iota_crio 6056   lecple 14250   occoc 14251   Atomscatm 32913   LHypclh 33633   LTrncltrn 33750   TEndoctendo 34401   DIsoCcdic 34822
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-dic 34823
This theorem is referenced by:  dicval  34826  dicfnN  34833
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