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Theorem dibfval 31257
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
dibfval  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) )
Distinct variable groups:    x, f, K    x, J    f, W, x
Allowed substitution hints:    B( x, f)    T( x, f)    H( x, f)    I( x, f)    J( f)    V( x, f)    .0. ( x, f)

Proof of Theorem dibfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dibval.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
2 dibval.b . . . . 5  |-  B  =  ( Base `  K
)
3 dibval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3dibffval 31256 . . . 4  |-  ( K  e.  V  ->  ( DIsoB `  K )  =  ( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K
) `  w )  |->  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) ) )
54fveq1d 5671 . . 3  |-  ( K  e.  V  ->  (
( DIsoB `  K ) `  W )  =  ( ( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K
) `  w )  |->  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) ) `  W ) )
61, 5syl5eq 2432 . 2  |-  ( K  e.  V  ->  I  =  ( ( w  e.  H  |->  ( x  e.  dom  ( (
DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } ) ) ) `  W ) )
7 fveq2 5669 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
8 dibval.j . . . . . 6  |-  J  =  ( ( DIsoA `  K
) `  W )
97, 8syl6eqr 2438 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  J )
109dmeqd 5013 . . . 4  |-  ( w  =  W  ->  dom  ( ( DIsoA `  K
) `  w )  =  dom  J )
119fveq1d 5671 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoA `  K
) `  w ) `  x )  =  ( J `  x ) )
12 fveq2 5669 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
13 dibval.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1412, 13syl6eqr 2438 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
15 eqidd 2389 . . . . . . . 8  |-  ( w  =  W  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
1614, 15mpteq12dv 4229 . . . . . . 7  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) )  =  ( f  e.  T  |->  (  _I  |`  B ) ) )
17 dibval.o . . . . . . 7  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
1816, 17syl6eqr 2438 . . . . . 6  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) )  =  .0.  )
1918sneqd 3771 . . . . 5  |-  ( w  =  W  ->  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) }  =  {  .0.  }
)
2011, 19xpeq12d 4844 . . . 4  |-  ( w  =  W  ->  (
( ( ( DIsoA `  K ) `  w
) `  x )  X.  { ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } )  =  ( ( J `  x )  X.  {  .0.  } ) )
2110, 20mpteq12dv 4229 . . 3  |-  ( w  =  W  ->  (
x  e.  dom  (
( DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } ) )  =  ( x  e. 
dom  J  |->  ( ( J `  x )  X.  {  .0.  }
) ) )
22 eqid 2388 . . 3  |-  ( w  e.  H  |->  ( x  e.  dom  ( (
DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } ) ) )  =  ( w  e.  H  |->  ( x  e.  dom  ( (
DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K
) `  w ) `  x )  X.  {
( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } ) ) )
23 fvex 5683 . . . . . 6  |-  ( (
DIsoA `  K ) `  W )  e.  _V
248, 23eqeltri 2458 . . . . 5  |-  J  e. 
_V
2524dmex 5073 . . . 4  |-  dom  J  e.  _V
2625mptex 5906 . . 3  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  e.  _V
2721, 22, 26fvmpt 5746 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K
) `  w )  |->  ( ( ( (
DIsoA `  K ) `  w ) `  x
)  X.  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) } ) ) ) `  W )  =  ( x  e.  dom  J  |->  ( ( J `  x )  X.  {  .0.  } ) ) )
286, 27sylan9eq 2440 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758    e. cmpt 4208    _I cid 4435    X. cxp 4817   dom cdm 4819    |` cres 4821   ` cfv 5395   Basecbs 13397   LHypclh 30099   LTrncltrn 30216   DIsoAcdia 31144   DIsoBcdib 31254
This theorem is referenced by:  dibval  31258  dibfna  31270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-dib 31255
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