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Theorem dicval2 31428
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 31425 . 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 5635 . . . . 5  |-  ( s `
 G )  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )
1110eqeq2i 2376 . . . 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 4185 . 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 2416 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 1647    e. wcel 1715   class class class wbr 4125   {copab 4178   ` cfv 5358   iota_crio 6439   lecple 13423   occoc 13424   Atomscatm 29512   LHypclh 30232   LTrncltrn 30349   TEndoctendo 31000   DIsoCcdic 31421
This theorem is referenced by:  dicelval3  31429  diclspsn  31443  dih1dimatlem  31578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6446  df-dic 31422
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