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Theorem dicelval1sta 31999
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.)
Hypotheses
Ref Expression
dicelval1sta.l  |-  .<_  =  ( le `  K )
dicelval1sta.a  |-  A  =  ( Atoms `  K )
dicelval1sta.h  |-  H  =  ( LHyp `  K
)
dicelval1sta.p  |-  P  =  ( ( oc `  K ) `  W
)
dicelval1sta.t  |-  T  =  ( ( LTrn `  K
) `  W )
dicelval1sta.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicelval1sta  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
Distinct variable groups:    g, K    Q, g    T, g    g, W
Allowed substitution hints:    A( g)    P( g)    H( g)    I( g)    .<_ ( g)    V( g)    Y( g)

Proof of Theorem dicelval1sta
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicelval1sta.l . . . . . 6  |-  .<_  =  ( le `  K )
2 dicelval1sta.a . . . . . 6  |-  A  =  ( Atoms `  K )
3 dicelval1sta.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 dicelval1sta.p . . . . . 6  |-  P  =  ( ( oc `  K ) `  W
)
5 dicelval1sta.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
6 eqid 2296 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dicelval1sta.i . . . . . 6  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicval 31988 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
98eleq2d 2363 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } ) )
109biimp3a 1281 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  Y  e.  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
11 eqeq1 2302 . . . . 5  |-  ( f  =  ( 1st `  Y
)  ->  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  <->  ( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
1211anbi1d 685 . . . 4  |-  ( f  =  ( 1st `  Y
)  ->  ( (
f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  ( ( 1st `  Y )  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) ) )
13 fveq1 5540 . . . . . 6  |-  ( s  =  ( 2nd `  Y
)  ->  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
1413eqeq2d 2307 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( ( 1st `  Y )  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  <-> 
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
15 eleq1 2356 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( s  e.  ( ( TEndo `  K
) `  W )  <->  ( 2nd `  Y )  e.  ( ( TEndo `  K ) `  W
) ) )
1614, 15anbi12d 691 . . . 4  |-  ( s  =  ( 2nd `  Y
)  ->  ( (
( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
) ) )
1712, 16elopabi 6201 . . 3  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  ->  (
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
) )
1810, 17syl 15 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  (
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
) )
1918simpld 445 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   {copab 4092   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   lecple 13231   occoc 13232   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DIsoCcdic 31984
This theorem is referenced by:  dicvaddcl  32002  dicvscacl  32003
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-pw 3640  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-1st 6138  df-2nd 6139  df-riota 6320  df-dic 31985
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