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Theorem diaval 30489
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
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 30488 . . . 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 5487 . 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 4027 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  W  <->  X  .<_  W ) )
1211elrab 2924 . . . 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 4028 . . . . 5  |-  ( x  =  X  ->  (
( R `  f
)  .<_  x  <->  ( R `  f )  .<_  X ) )
1514rabbidv 2781 . . . 4  |-  ( x  =  X  ->  { f  e.  T  |  ( R `  f ) 
.<_  x }  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
16 eqid 2284 . . . 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 5499 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
184, 17eqeltri 2354 . . . . 5  |-  T  e. 
_V
1918rabex 4166 . . . 4  |-  { f  e.  T  |  ( R `  f ) 
.<_  X }  e.  _V
2015, 16, 19fvmpt 5563 . . 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 2316 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 1624    e. wcel 1685   {crab 2548   _Vcvv 2789   class class class wbr 4024    e. cmpt 4078   ` cfv 5221   Basecbs 13142   lecple 13209   LHypclh 29440   LTrncltrn 29557   trLctrl 29614   DIsoAcdia 30485
This theorem is referenced by:  diaelval  30490  diass  30499  diaord  30504  dia0  30509  dia1N  30510  diassdvaN  30517  dia1dim  30518  cdlemm10N  30575  dibval3N  30603  dihwN  30746
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-disoa 30486
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