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Theorem dibfval 31331
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 31330 . . . 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 5527 . . 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 2327 . 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 5525 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
8 dibval.j . . . . . 6  |-  J  =  ( ( DIsoA `  K
) `  W )
97, 8syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  J )
109dmeqd 4881 . . . 4  |-  ( w  =  W  ->  dom  ( ( DIsoA `  K
) `  w )  =  dom  J )
119fveq1d 5527 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoA `  K
) `  w ) `  x )  =  ( J `  x ) )
12 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
13 dibval.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1412, 13syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
15 eqidd 2284 . . . . . . . 8  |-  ( w  =  W  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
1614, 15mpteq12dv 4098 . . . . . . 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 2333 . . . . . 6  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) )  =  .0.  )
1918sneqd 3653 . . . . 5  |-  ( w  =  W  ->  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) }  =  {  .0.  }
)
2011, 19xpeq12d 4714 . . . 4  |-  ( w  =  W  ->  (
( ( ( DIsoA `  K ) `  w
) `  x )  X.  { ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } )  =  ( ( J `  x )  X.  {  .0.  } ) )
2110, 20mpteq12dv 4098 . . 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 2283 . . 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 5539 . . . . . 6  |-  ( (
DIsoA `  K ) `  W )  e.  _V
248, 23eqeltri 2353 . . . . 5  |-  J  e. 
_V
2524dmex 4941 . . . 4  |-  dom  J  e.  _V
2625mptex 5746 . . 3  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  e.  _V
2721, 22, 26fvmpt 5602 . 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 2335 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689    |` cres 4691   ` cfv 5255   Basecbs 13148   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibval  31332  dibfna  31344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dib 31329
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