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Theorem dibval 31941
Description: The partial isomorphism B for a lattice  K. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
dibval.b  |-  B  =  ( Base `  K
)
dibval.h  |-  H  =  ( LHyp `  K
)
dibval.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibval.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
dibval.j  |-  J  =  ( ( DIsoA `  K
) `  W )
dibval.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Distinct variable groups:    f, K    f, W
Allowed substitution hints:    B( f)    T( f)    H( f)    I( f)    J( f)    V( f)    X( f)    .0. ( f)

Proof of Theorem dibval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dibval.b . . . . 5  |-  B  =  ( Base `  K
)
2 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 dibval.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
4 dibval.o . . . . 5  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
5 dibval.j . . . . 5  |-  J  =  ( ( DIsoA `  K
) `  W )
6 dibval.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
71, 2, 3, 4, 5, 6dibfval 31940 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) )
87adantr 453 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  I  =  ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) )
98fveq1d 5731 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) `  X
) )
10 fveq2 5729 . . . . 5  |-  ( x  =  X  ->  ( J `  x )  =  ( J `  X ) )
1110xpeq1d 4902 . . . 4  |-  ( x  =  X  ->  (
( J `  x
)  X.  {  .0.  } )  =  ( ( J `  X )  X.  {  .0.  }
) )
12 eqid 2437 . . . 4  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  =  ( x  e.  dom  J  |->  ( ( J `  x )  X.  {  .0.  } ) )
13 fvex 5743 . . . . 5  |-  ( J `
 X )  e. 
_V
14 snex 4406 . . . . 5  |-  {  .0.  }  e.  _V
1513, 14xpex 4991 . . . 4  |-  ( ( J `  X )  X.  {  .0.  }
)  e.  _V
1611, 12, 15fvmpt 5807 . . 3  |-  ( X  e.  dom  J  -> 
( ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) `  X
)  =  ( ( J `  X )  X.  {  .0.  }
) )
1716adantl 454 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
( x  e.  dom  J 
|->  ( ( J `  x )  X.  {  .0.  } ) ) `  X )  =  ( ( J `  X
)  X.  {  .0.  } ) )
189, 17eqtrd 2469 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {csn 3815    e. cmpt 4267    _I cid 4494    X. cxp 4877   dom cdm 4879    |` cres 4881   ` cfv 5455   Basecbs 13470   LHypclh 30782   LTrncltrn 30899   DIsoAcdia 31827   DIsoBcdib 31937
This theorem is referenced by:  dibopelvalN  31942  dibval2  31943  dibvalrel  31962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-dib 31938
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