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Theorem dicopelval2 35165
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 35161 . 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 5803 . . . 4  |-  ( S `
 G )  =  ( S `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )
1312eqeq2i 2472 . . 3  |-  ( F  =  ( S `  G )  <->  F  =  ( S `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) ) )
1413anbi1i 695 . 2  |-  ( ( F  =  ( S `
 G )  /\  S  e.  E )  <->  ( F  =  ( S `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  S  e.  E ) )
1510, 14syl6bbr 263 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3992   class class class wbr 4401   ` cfv 5527   iota_crio 6161   lecple 14365   occoc 14366   Atomscatm 33247   LHypclh 33967   LTrncltrn 34084   TEndoctendo 34735   DIsoCcdic 35156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-dic 35157
This theorem is referenced by:  diclspsn  35178  cdlemn11a  35191  dihopelvalcqat  35230  dihopelvalcpre  35232  dihord6apre  35240
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