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Theorem dibval 37012
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 37011 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) )
87adantr 465 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  I  =  ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) )
98fveq1d 5874 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) `  X
) )
10 fveq2 5872 . . . . 5  |-  ( x  =  X  ->  ( J `  x )  =  ( J `  X ) )
1110xpeq1d 5031 . . . 4  |-  ( x  =  X  ->  (
( J `  x
)  X.  {  .0.  } )  =  ( ( J `  X )  X.  {  .0.  }
) )
12 eqid 2457 . . . 4  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  =  ( x  e.  dom  J  |->  ( ( J `  x )  X.  {  .0.  } ) )
13 fvex 5882 . . . . 5  |-  ( J `
 X )  e. 
_V
14 snex 4697 . . . . 5  |-  {  .0.  }  e.  _V
1513, 14xpex 6603 . . . 4  |-  ( ( J `  X )  X.  {  .0.  }
)  e.  _V
1611, 12, 15fvmpt 5956 . . 3  |-  ( X  e.  dom  J  -> 
( ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) `  X
)  =  ( ( J `  X )  X.  {  .0.  }
) )
1716adantl 466 . 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 2498 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {csn 4032    |-> cmpt 4515    _I cid 4799    X. cxp 5006   dom cdm 5008    |` cres 5010   ` cfv 5594   Basecbs 14644   LHypclh 35851   LTrncltrn 35968   DIsoAcdia 36898   DIsoBcdib 37008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-dib 37009
This theorem is referenced by:  dibopelvalN  37013  dibval2  37014  dibvalrel  37033
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