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Theorem diaval 31222
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 31221 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
87adantr 451 . . 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 5527 . 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 447 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  e.  B  /\  X  .<_  W ) )
11 breq1 4026 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  W  <->  X  .<_  W ) )
1211elrab 2923 . . . 4  |-  ( X  e.  { y  e.  B  |  y  .<_  W }  <->  ( X  e.  B  /\  X  .<_  W ) )
1310, 12sylibr 203 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  e.  { y  e.  B  |  y  .<_  W }
)
14 breq2 4027 . . . . 5  |-  ( x  =  X  ->  (
( R `  f
)  .<_  x  <->  ( R `  f )  .<_  X ) )
1514rabbidv 2780 . . . 4  |-  ( x  =  X  ->  { f  e.  T  |  ( R `  f ) 
.<_  x }  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
16 eqid 2283 . . . 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 5539 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
184, 17eqeltri 2353 . . . . 5  |-  T  e. 
_V
1918rabex 4165 . . . 4  |-  { f  e.  T  |  ( R `  f ) 
.<_  X }  e.  _V
2015, 16, 19fvmpt 5602 . . 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 15 . 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 2315 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 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   DIsoAcdia 31218
This theorem is referenced by:  diaelval  31223  diass  31232  diaord  31237  dia0  31242  dia1N  31243  diassdvaN  31250  dia1dim  31251  cdlemm10N  31308  dibval3N  31336  dihwN  31479
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-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
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-disoa 31219
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