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Theorem dicelval1sta 34830
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 2442 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dicelval1sta.i . . . . . 6  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicval 34819 . . . . 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 2509 . . . 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 1318 . . 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 2448 . . . . 5  |-  ( f  =  ( 1st `  Y
)  ->  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  <->  ( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
1211anbi1d 704 . . . 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 5689 . . . . . 6  |-  ( s  =  ( 2nd `  Y
)  ->  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) ) )
1413eqeq2d 2453 . . . . 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 2502 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( s  e.  ( ( TEndo `  K
) `  W )  <->  ( 2nd `  Y )  e.  ( ( TEndo `  K ) `  W
) ) )
1614, 15anbi12d 710 . . . 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 6634 . . 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 16 . 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 459 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4291   {copab 4348   ` cfv 5417   iota_crio 6050   1stc1st 6574   2ndc2nd 6575   lecple 14244   occoc 14245   Atomscatm 32906   LHypclh 33626   LTrncltrn 33743   TEndoctendo 34394   DIsoCcdic 34815
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-1st 6576  df-2nd 6577  df-dic 34816
This theorem is referenced by:  dicvaddcl  34833  dicvscacl  34834
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