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Theorem dicfval 31987
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 31986 . . . 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 5543 . . 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 2340 . 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 4043 . . . . . 6  |-  ( w  =  W  ->  (
r  .<_  w  <->  r  .<_  W ) )
98notbid 285 . . . . 5  |-  ( w  =  W  ->  ( -.  r  .<_  w  <->  -.  r  .<_  W ) )
109rabbidv 2793 . . . 4  |-  ( w  =  W  ->  { r  e.  A  |  -.  r  .<_  w }  =  { r  e.  A  |  -.  r  .<_  W }
)
11 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
12 dicval.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
1311, 12syl6eqr 2346 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
14 fveq2 5541 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( oc `  K
) `  w )  =  ( ( oc
`  K ) `  W ) )
15 dicval.p . . . . . . . . . . . 12  |-  P  =  ( ( oc `  K ) `  W
)
1614, 15syl6eqr 2346 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( oc `  K
) `  w )  =  P )
1716fveq2d 5545 . . . . . . . . . 10  |-  ( w  =  W  ->  (
g `  ( ( oc `  K ) `  w ) )  =  ( g `  P
) )
1817eqeq1d 2304 . . . . . . . . 9  |-  ( w  =  W  ->  (
( g `  (
( oc `  K
) `  w )
)  =  q  <->  ( g `  P )  =  q ) )
1913, 18riotaeqbidv 6323 . . . . . . . 8  |-  ( w  =  W  ->  ( iota_ g  e.  ( (
LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q )  =  (
iota_ g  e.  T
( g `  P
)  =  q ) )
2019fveq2d 5545 . . . . . . 7  |-  ( w  =  W  ->  (
s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  q ) ) )
2120eqeq2d 2307 . . . . . 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 5541 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
23 dicval.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
2422, 23syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
2524eleq2d 2363 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w )  <->  s  e.  E ) )
2621, 25anbi12d 691 . . . . 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 4098 . . . 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 4114 . . 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 2296 . . 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 5555 . . . . . 6  |-  ( Atoms `  K )  e.  _V
313, 30eqeltri 2366 . . . . 5  |-  A  e. 
_V
3231rabex 4181 . . . 4  |-  { r  e.  A  |  -.  r  .<_  W }  e.  _V
3332mptex 5762 . . 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 5618 . 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 2348 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039   {copab 4092    e. cmpt 4093   ` cfv 5271   iota_crio 6313   lecple 13231   occoc 13232   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DIsoCcdic 31984
This theorem is referenced by:  dicval  31988  dicfnN  31995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-dic 31985
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