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Theorem diaval 31515
Description: The partial isomorphism A for a lattice  K. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b  |-  B  =  ( Base `  K
)
diaval.l  |-  .<_  =  ( le `  K )
diaval.h  |-  H  =  ( LHyp `  K
)
diaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
diaval.r  |-  R  =  ( ( trL `  K
) `  W )
diaval.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( R `  f )  .<_  X }
)
Distinct variable groups:    f, K    T, f    f, W    f, X
Allowed substitution hints:    B( f)    R( f)    H( f)    I( f)    .<_ ( f)    V( f)

Proof of Theorem diaval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . . 5  |-  B  =  ( Base `  K
)
2 diaval.l . . . . 5  |-  .<_  =  ( le `  K )
3 diaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 diaval.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
5 diaval.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
6 diaval.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diafval 31514 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
87adantr 452 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  I  =  ( x  e. 
{ y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f )  .<_  x }
) )
98fveq1d 5689 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) `  X ) )
10 simpr 448 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  e.  B  /\  X  .<_  W ) )
11 breq1 4175 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  W  <->  X  .<_  W ) )
1211elrab 3052 . . . 4  |-  ( X  e.  { y  e.  B  |  y  .<_  W }  <->  ( X  e.  B  /\  X  .<_  W ) )
1310, 12sylibr 204 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  e.  { y  e.  B  |  y  .<_  W }
)
14 breq2 4176 . . . . 5  |-  ( x  =  X  ->  (
( R `  f
)  .<_  x  <->  ( R `  f )  .<_  X ) )
1514rabbidv 2908 . . . 4  |-  ( x  =  X  ->  { f  e.  T  |  ( R `  f ) 
.<_  x }  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
16 eqid 2404 . . . 4  |-  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } )  =  ( x  e.  {
y  e.  B  | 
y  .<_  W }  |->  { f  e.  T  | 
( R `  f
)  .<_  x } )
17 fvex 5701 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
184, 17eqeltri 2474 . . . . 5  |-  T  e. 
_V
1918rabex 4314 . . . 4  |-  { f  e.  T  |  ( R `  f ) 
.<_  X }  e.  _V
2015, 16, 19fvmpt 5765 . . 3  |-  ( X  e.  { y  e.  B  |  y  .<_  W }  ->  ( ( x  e.  { y  e.  B  |  y 
.<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) `  X )  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
2113, 20syl 16 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( x  e.  {
y  e.  B  | 
y  .<_  W }  |->  { f  e.  T  | 
( R `  f
)  .<_  x } ) `
 X )  =  { f  e.  T  |  ( R `  f )  .<_  X }
)
229, 21eqtrd 2436 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( R `  f )  .<_  X }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413   Basecbs 13424   lecple 13491   LHypclh 30466   LTrncltrn 30583   trLctrl 30640   DIsoAcdia 31511
This theorem is referenced by:  diaelval  31516  diass  31525  diaord  31530  dia0  31535  dia1N  31536  diassdvaN  31543  dia1dim  31544  cdlemm10N  31601  dibval3N  31629  dihwN  31772
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-disoa 31512
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