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Theorem dibfval 31876
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 31875 . . . 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 5722 . . 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 2479 . 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 5720 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
8 dibval.j . . . . . 6  |-  J  =  ( ( DIsoA `  K
) `  W )
97, 8syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  J )
109dmeqd 5064 . . . 4  |-  ( w  =  W  ->  dom  ( ( DIsoA `  K
) `  w )  =  dom  J )
119fveq1d 5722 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoA `  K
) `  w ) `  x )  =  ( J `  x ) )
12 fveq2 5720 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
13 dibval.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1412, 13syl6eqr 2485 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
15 eqidd 2436 . . . . . . . 8  |-  ( w  =  W  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
1614, 15mpteq12dv 4279 . . . . . . 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 2485 . . . . . 6  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) )  =  .0.  )
1918sneqd 3819 . . . . 5  |-  ( w  =  W  ->  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) }  =  {  .0.  }
)
2011, 19xpeq12d 4895 . . . 4  |-  ( w  =  W  ->  (
( ( ( DIsoA `  K ) `  w
) `  x )  X.  { ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } )  =  ( ( J `  x )  X.  {  .0.  } ) )
2110, 20mpteq12dv 4279 . . 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 2435 . . 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 5734 . . . . . 6  |-  ( (
DIsoA `  K ) `  W )  e.  _V
248, 23eqeltri 2505 . . . . 5  |-  J  e. 
_V
2524dmex 5124 . . . 4  |-  dom  J  e.  _V
2625mptex 5958 . . 3  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  e.  _V
2721, 22, 26fvmpt 5798 . 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 2487 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 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    e. cmpt 4258    _I cid 4485    X. cxp 4868   dom cdm 4870    |` cres 4872   ` cfv 5446   Basecbs 13461   LHypclh 30718   LTrncltrn 30835   DIsoAcdia 31763   DIsoBcdib 31873
This theorem is referenced by:  dibval  31877  dibfna  31889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-dib 31874
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