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Theorem pmapval 30568
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 30567 . . 3  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
65fveq1d 5543 . 2  |-  ( K  e.  C  ->  ( M `  X )  =  ( ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } ) `  X ) )
7 breq2 4043 . . . 4  |-  ( x  =  X  ->  (
a  .<_  x  <->  a  .<_  X ) )
87rabbidv 2793 . . 3  |-  ( x  =  X  ->  { a  e.  A  |  a 
.<_  x }  =  {
a  e.  A  | 
a  .<_  X } )
9 eqid 2296 . . 3  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
)
10 fvex 5555 . . . . 5  |-  ( Atoms `  K )  e.  _V
113, 10eqeltri 2366 . . . 4  |-  A  e. 
_V
1211rabex 4181 . . 3  |-  { a  e.  A  |  a 
.<_  X }  e.  _V
138, 9, 12fvmpt 5618 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) `  X )  =  { a  e.  A  |  a  .<_  X }
)
146, 13sylan9eq 2348 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { a  e.  A  |  a 
.<_  X } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   Basecbs 13164   lecple 13231   Atomscatm 30075   pmapcpmap 30308
This theorem is referenced by:  elpmap  30569  pmapssat  30570  pmaple  30572  pmapat  30574  pmap0  30576  pmap1N  30578  pmapsub  30579  pmapglbx  30580  isline2  30585  linepmap  30586  polpmapN  30723  2polssN  30726  pmaplubN  30735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-pmap 30315
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