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Theorem dicval2 31369
Description: 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 )
Assertion
Ref Expression
dicval2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } )
Distinct variable groups:    f, g,
s, K    T, g    f, W, g, s    f, E, s    P, f    Q, f, g, s    T, f
Allowed substitution hints:    A( f, g, s)    P( g, s)    T( s)    E( g)    G( f, g, s)    H( f, g, s)    I( f, g, s)    .<_ ( f, g, s)    V( f, g, s)

Proof of Theorem dicval2
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, 7dicval 31366 . 2  |-  ( ( ( 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.  E ) } )
9 dicval2.g . . . . . 6  |-  G  =  ( iota_ g  e.  T
( g `  P
)  =  Q )
109fveq2i 5528 . . . . 5  |-  ( s `
 G )  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )
1110eqeq2i 2293 . . . 4  |-  ( f  =  ( s `  G )  <->  f  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
1211anbi1i 676 . . 3  |-  ( ( f  =  ( s `
 G )  /\  s  e.  E )  <->  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) )
1312opabbii 4083 . 2  |-  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) }  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }
148, 13syl6eqr 2333 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   {copab 4076   ` cfv 5255   iota_crio 6297   lecple 13215   occoc 13216   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DIsoCcdic 31362
This theorem is referenced by:  dicelval3  31370  diclspsn  31384  dih1dimatlem  31519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304  df-dic 31363
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