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Theorem dibval 31954
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 31953 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) )
87adantr 451 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  I  =  ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) )
98fveq1d 5543 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) `  X
) )
10 fveq2 5541 . . . . 5  |-  ( x  =  X  ->  ( J `  x )  =  ( J `  X ) )
1110xpeq1d 4728 . . . 4  |-  ( x  =  X  ->  (
( J `  x
)  X.  {  .0.  } )  =  ( ( J `  X )  X.  {  .0.  }
) )
12 eqid 2296 . . . 4  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  =  ( x  e.  dom  J  |->  ( ( J `  x )  X.  {  .0.  } ) )
13 fvex 5555 . . . . 5  |-  ( J `
 X )  e. 
_V
14 snex 4232 . . . . 5  |-  {  .0.  }  e.  _V
1513, 14xpex 4817 . . . 4  |-  ( ( J `  X )  X.  {  .0.  }
)  e.  _V
1611, 12, 15fvmpt 5618 . . 3  |-  ( X  e.  dom  J  -> 
( ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) `  X
)  =  ( ( J `  X )  X.  {  .0.  }
) )
1716adantl 452 . 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 2328 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 358    = wceq 1632    e. wcel 1696   {csn 3653    e. cmpt 4093    _I cid 4320    X. cxp 4703   dom cdm 4705    |` cres 4707   ` cfv 5271   Basecbs 13164   LHypclh 30795   LTrncltrn 30912   DIsoAcdia 31840   DIsoBcdib 31950
This theorem is referenced by:  dibopelvalN  31955  dibval2  31956  dibvalrel  31975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-dib 31951
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