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Theorem diaval 29911
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 29910 . . . 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 5379 . 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 3923 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  W  <->  X  .<_  W ) )
1211elrab 2860 . . . 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 3924 . . . . 5  |-  ( x  =  X  ->  (
( R `  f
)  .<_  x  <->  ( R `  f )  .<_  X ) )
1514rabbidv 2719 . . . 4  |-  ( x  =  X  ->  { f  e.  T  |  ( R `  f ) 
.<_  x }  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
16 eqid 2253 . . . 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 5391 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
184, 17eqeltri 2323 . . . . 5  |-  T  e. 
_V
1918rabex 4061 . . . 4  |-  { f  e.  T  |  ( R `  f ) 
.<_  X }  e.  _V
2015, 16, 19fvmpt 5454 . . 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 2285 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 2512   _Vcvv 2727   class class class wbr 3920    e. cmpt 3974   ` cfv 4592   Basecbs 13022   lecple 13089   LHypclh 28862   LTrncltrn 28979   trLctrl 29036   DIsoAcdia 29907
This theorem is referenced by:  diaelval  29912  diass  29921  diaord  29926  dia0  29931  dia1N  29932  diassdvaN  29939  dia1dim  29940  cdlemm10N  29997  dibval3N  30025  dihwN  30168
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-disoa 29908
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