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Theorem diaval 30373
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
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 30372 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
87adantr 453 . . 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 5446 . 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 449 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  e.  B  /\  X  .<_  W ) )
11 breq1 3986 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  W  <->  X  .<_  W ) )
1211elrab 2891 . . . 4  |-  ( X  e.  { y  e.  B  |  y  .<_  W }  <->  ( X  e.  B  /\  X  .<_  W ) )
1310, 12sylibr 205 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  e.  { y  e.  B  |  y  .<_  W }
)
14 breq2 3987 . . . . 5  |-  ( x  =  X  ->  (
( R `  f
)  .<_  x  <->  ( R `  f )  .<_  X ) )
1514rabbidv 2749 . . . 4  |-  ( x  =  X  ->  { f  e.  T  |  ( R `  f ) 
.<_  x }  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
16 eqid 2256 . . . 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 5458 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
184, 17eqeltri 2326 . . . . 5  |-  T  e. 
_V
1918rabex 4125 . . . 4  |-  { f  e.  T  |  ( R `  f ) 
.<_  X }  e.  _V
2015, 16, 19fvmpt 5522 . . 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 17 . 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 2288 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   {crab 2520   _Vcvv 2757   class class class wbr 3983    e. cmpt 4037   ` cfv 4659   Basecbs 13096   lecple 13163   LHypclh 29324   LTrncltrn 29441   trLctrl 29498   DIsoAcdia 30369
This theorem is referenced by:  diaelval  30374  diass  30383  diaord  30388  dia0  30393  dia1N  30394  diassdvaN  30401  dia1dim  30402  cdlemm10N  30459  dibval3N  30487  dihwN  30630
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-disoa 30370
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