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Theorem pmapval 34770
Description: Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
pmapfval.b  |-  B  =  ( Base `  K
)
pmapfval.l  |-  .<_  =  ( le `  K )
pmapfval.a  |-  A  =  ( Atoms `  K )
pmapfval.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapval  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { a  e.  A  |  a 
.<_  X } )
Distinct variable groups:    A, a    K, a    X, a
Allowed substitution hints:    B( a)    C( a)   
.<_ ( a)    M( a)

Proof of Theorem pmapval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pmapfval.b . . . 4  |-  B  =  ( Base `  K
)
2 pmapfval.l . . . 4  |-  .<_  =  ( le `  K )
3 pmapfval.a . . . 4  |-  A  =  ( Atoms `  K )
4 pmapfval.m . . . 4  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapfval 34769 . . 3  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
65fveq1d 5868 . 2  |-  ( K  e.  C  ->  ( M `  X )  =  ( ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } ) `  X ) )
7 breq2 4451 . . . 4  |-  ( x  =  X  ->  (
a  .<_  x  <->  a  .<_  X ) )
87rabbidv 3105 . . 3  |-  ( x  =  X  ->  { a  e.  A  |  a 
.<_  x }  =  {
a  e.  A  | 
a  .<_  X } )
9 eqid 2467 . . 3  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
)
10 fvex 5876 . . . . 5  |-  ( Atoms `  K )  e.  _V
113, 10eqeltri 2551 . . . 4  |-  A  e. 
_V
1211rabex 4598 . . 3  |-  { a  e.  A  |  a 
.<_  X }  e.  _V
138, 9, 12fvmpt 5951 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) `  X )  =  { a  e.  A  |  a  .<_  X }
)
146, 13sylan9eq 2528 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { a  e.  A  |  a 
.<_  X } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588   Basecbs 14493   lecple 14565   Atomscatm 34277   pmapcpmap 34510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-pmap 34517
This theorem is referenced by:  elpmap  34771  pmapssat  34772  pmaple  34774  pmapat  34776  pmap0  34778  pmap1N  34780  pmapsub  34781  pmapglbx  34782  isline2  34787  linepmap  34788  polpmapN  34925  2polssN  34928  pmaplubN  34937
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