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Theorem dibfval 36608
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 36607 . . . 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 5858 . . 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 2496 . 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 5856 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  ( ( DIsoA `  K ) `  W ) )
8 dibval.j . . . . . 6  |-  J  =  ( ( DIsoA `  K
) `  W )
97, 8syl6eqr 2502 . . . . 5  |-  ( w  =  W  ->  (
( DIsoA `  K ) `  w )  =  J )
109dmeqd 5195 . . . 4  |-  ( w  =  W  ->  dom  ( ( DIsoA `  K
) `  w )  =  dom  J )
119fveq1d 5858 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoA `  K
) `  w ) `  x )  =  ( J `  x ) )
12 fveq2 5856 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
13 dibval.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
1412, 13syl6eqr 2502 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
15 eqidd 2444 . . . . . . . 8  |-  ( w  =  W  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
1614, 15mpteq12dv 4515 . . . . . . 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 2502 . . . . . 6  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) )  =  .0.  )
1918sneqd 4026 . . . . 5  |-  ( w  =  W  ->  { ( f  e.  ( (
LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) }  =  {  .0.  }
)
2011, 19xpeq12d 5014 . . . 4  |-  ( w  =  W  ->  (
( ( ( DIsoA `  K ) `  w
) `  x )  X.  { ( f  e.  ( ( LTrn `  K
) `  w )  |->  (  _I  |`  B ) ) } )  =  ( ( J `  x )  X.  {  .0.  } ) )
2110, 20mpteq12dv 4515 . . 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 2443 . . 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 5866 . . . . . 6  |-  ( (
DIsoA `  K ) `  W )  e.  _V
248, 23eqeltri 2527 . . . . 5  |-  J  e. 
_V
2524dmex 6718 . . . 4  |-  dom  J  e.  _V
2625mptex 6128 . . 3  |-  ( x  e.  dom  J  |->  ( ( J `  x
)  X.  {  .0.  } ) )  e.  _V
2721, 22, 26fvmpt 5941 . 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 2504 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 1383    e. wcel 1804   _Vcvv 3095   {csn 4014    |-> cmpt 4495    _I cid 4780    X. cxp 4987   dom cdm 4989    |` cres 4991   ` cfv 5578   Basecbs 14509   LHypclh 35448   LTrncltrn 35565   DIsoAcdia 36495   DIsoBcdib 36605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-dib 36606
This theorem is referenced by:  dibval  36609  dibfna  36621
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