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Theorem dicelval1sta 34464
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 2429 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dicelval1sta.i . . . . . 6  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicval 34453 . . . . 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 2499 . . . 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 1364 . . 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 2433 . . . . 5  |-  ( f  =  ( 1st `  Y
)  ->  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  <->  ( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
1211anbi1d 709 . . . 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 5880 . . . . . 6  |-  ( s  =  ( 2nd `  Y
)  ->  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) ) )
1413eqeq2d 2443 . . . . 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 2501 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( s  e.  ( ( TEndo `  K
) `  W )  <->  ( 2nd `  Y )  e.  ( ( TEndo `  K ) `  W
) ) )
1614, 15anbi12d 715 . . . 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 6868 . . 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 17 . 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 460 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   {copab 4483   ` cfv 5601   iota_crio 6266   1stc1st 6805   2ndc2nd 6806   lecple 15159   occoc 15160   Atomscatm 32538   LHypclh 33258   LTrncltrn 33375   TEndoctendo 34028   DIsoCcdic 34449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-1st 6807  df-2nd 6808  df-dic 34450
This theorem is referenced by:  dicvaddcl  34467  dicvscacl  34468
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