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Theorem dicelval2N 31917
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
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 )
Assertion
Ref Expression
dicelval2N  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 G )  /\  ( 2nd `  Y )  e.  E ) ) ) )
Distinct variable groups:    g, K    T, g    g, W    Q, g
Allowed substitution hints:    A( g)    P( g)    E( g)    G( g)    H( g)    I( g)    .<_ ( g)    V( g)    Y( g)

Proof of Theorem dicelval2N
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 )
81, 2, 3, 4, 5, 6, 7dicelvalN 31913 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) ) )
9 dicval2.g . . . . . 6  |-  G  =  ( iota_ g  e.  T
( g `  P
)  =  Q )
109fveq2i 5723 . . . . 5  |-  ( ( 2nd `  Y ) `
 G )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )
1110eqeq2i 2445 . . . 4  |-  ( ( 1st `  Y )  =  ( ( 2nd `  Y ) `  G
)  <->  ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
1211anbi1i 677 . . 3  |-  ( ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 G )  /\  ( 2nd `  Y )  e.  E )  <->  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) )
1312anbi2i 676 . 2  |-  ( ( Y  e.  ( _V 
X.  _V )  /\  (
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 G )  /\  ( 2nd `  Y )  e.  E ) )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) )
148, 13syl6bbr 255 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 G )  /\  ( 2nd `  Y )  e.  E ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204    X. cxp 4868   ` cfv 5446   1stc1st 6339   2ndc2nd 6340   iota_crio 6534   lecple 13528   occoc 13529   Atomscatm 29998   LHypclh 30718   LTrncltrn 30835   TEndoctendo 31486   DIsoCcdic 31907
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1st 6341  df-2nd 6342  df-riota 6541  df-dic 31908
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