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Theorem dicopelval2 31979
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 20-Feb-2014.)
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 )
dicval2.g  |-  G  =  ( iota_ g  e.  T
( g `  P
)  =  Q )
dicelval2.f  |-  F  e. 
_V
dicelval2.s  |-  S  e. 
_V
Assertion
Ref Expression
dicopelval2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. F ,  S >.  e.  ( I `  Q )  <->  ( F  =  ( S `  G )  /\  S  e.  E ) ) )
Distinct variable groups:    g, K    T, g    g, W    Q, g
Allowed substitution hints:    A( g)    P( g)    S( g)    E( g)    F( g)    G( g)    H( g)    I( g)    .<_ ( g)    V( g)

Proof of Theorem dicopelval2
StepHypRef Expression
1 dicval.l . . 3  |-  .<_  =  ( le `  K )
2 dicval.a . . 3  |-  A  =  ( Atoms `  K )
3 dicval.h . . 3  |-  H  =  ( LHyp `  K
)
4 dicval.p . . 3  |-  P  =  ( ( oc `  K ) `  W
)
5 dicval.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
6 dicval.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
7 dicval.i . . 3  |-  I  =  ( ( DIsoC `  K
) `  W )
8 dicelval2.f . . 3  |-  F  e. 
_V
9 dicelval2.s . . 3  |-  S  e. 
_V
101, 2, 3, 4, 5, 6, 7, 8, 9dicopelval 31975 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. F ,  S >.  e.  ( I `  Q )  <->  ( F  =  ( S `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  S  e.  E ) ) )
11 dicval2.g . . . . 5  |-  G  =  ( iota_ g  e.  T
( g `  P
)  =  Q )
1211fveq2i 5731 . . . 4  |-  ( S `
 G )  =  ( S `  ( iota_ g  e.  T ( g `  P )  =  Q ) )
1312eqeq2i 2446 . . 3  |-  ( F  =  ( S `  G )  <->  F  =  ( S `  ( iota_ g  e.  T ( g `
 P )  =  Q ) ) )
1413anbi1i 677 . 2  |-  ( ( F  =  ( S `
 G )  /\  S  e.  E )  <->  ( F  =  ( S `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  S  e.  E ) )
1510, 14syl6bbr 255 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. F ,  S >.  e.  ( I `  Q )  <->  ( F  =  ( S `  G )  /\  S  e.  E ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212   ` cfv 5454   iota_crio 6542   lecple 13536   occoc 13537   Atomscatm 30061   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549   DIsoCcdic 31970
This theorem is referenced by:  diclspsn  31992  cdlemn11a  32005  dihopelvalcqat  32044  dihopelvalcpre  32046  dihord6apre  32054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-dic 31971
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