Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibfval Structured version   Unicode version

Theorem dibfval 34783
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 34782 . . . 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 5691 . . 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 2485 . 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 5689 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
8 dibval.j . . . . . 6  |-  J  =  ( ( DIsoA `  K
) `  W )
97, 8syl6eqr 2491 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  J )
109dmeqd 5040 . . . 4  |-  ( w  =  W  ->  dom  ( ( DIsoA `  K
) `  w )  =  dom  J )
119fveq1d 5691 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoA `  K
) `  w ) `  x )  =  ( J `  x ) )
12 fveq2 5689 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
13 dibval.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1412, 13syl6eqr 2491 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
15 eqidd 2442 . . . . . . . 8  |-  ( w  =  W  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
1614, 15mpteq12dv 4368 . . . . . . 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 2491 . . . . . 6  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) )  =  .0.  )
1918sneqd 3887 . . . . 5  |-  ( w  =  W  ->  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) }  =  {  .0.  }
)
2011, 19xpeq12d 4863 . . . 4  |-  ( w  =  W  ->  (
( ( ( DIsoA `  K ) `  w
) `  x )  X.  { ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } )  =  ( ( J `  x )  X.  {  .0.  } ) )
2110, 20mpteq12dv 4368 . . 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 2441 . . 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 5699 . . . . . 6  |-  ( (
DIsoA `  K ) `  W )  e.  _V
248, 23eqeltri 2511 . . . . 5  |-  J  e. 
_V
2524dmex 6509 . . . 4  |-  dom  J  e.  _V
2625mptex 5946 . . 3  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  e.  _V
2721, 22, 26fvmpt 5772 . 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 2493 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2970   {csn 3875    e. cmpt 4348    _I cid 4629    X. cxp 4836   dom cdm 4838    |` cres 4840   ` cfv 5416   Basecbs 14172   LHypclh 33625   LTrncltrn 33742   DIsoAcdia 34670   DIsoBcdib 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-dib 34781
This theorem is referenced by:  dibval  34784  dibfna  34796
  Copyright terms: Public domain W3C validator