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Theorem pmapval 30285
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 30284 . . 3  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
65fveq1d 5716 . 2  |-  ( K  e.  C  ->  ( M `  X )  =  ( ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } ) `  X ) )
7 breq2 4203 . . . 4  |-  ( x  =  X  ->  (
a  .<_  x  <->  a  .<_  X ) )
87rabbidv 2935 . . 3  |-  ( x  =  X  ->  { a  e.  A  |  a 
.<_  x }  =  {
a  e.  A  | 
a  .<_  X } )
9 eqid 2430 . . 3  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
)
10 fvex 5728 . . . . 5  |-  ( Atoms `  K )  e.  _V
113, 10eqeltri 2500 . . . 4  |-  A  e. 
_V
1211rabex 4341 . . 3  |-  { a  e.  A  |  a 
.<_  X }  e.  _V
138, 9, 12fvmpt 5792 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) `  X )  =  { a  e.  A  |  a  .<_  X }
)
146, 13sylan9eq 2482 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 359    = wceq 1652    e. wcel 1725   {crab 2696   _Vcvv 2943   class class class wbr 4199    e. cmpt 4253   ` cfv 5440   Basecbs 13452   lecple 13519   Atomscatm 29792   pmapcpmap 30025
This theorem is referenced by:  elpmap  30286  pmapssat  30287  pmaple  30289  pmapat  30291  pmap0  30293  pmap1N  30295  pmapsub  30296  pmapglbx  30297  isline2  30302  linepmap  30303  polpmapN  30440  2polssN  30443  pmaplubN  30452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-pmap 30032
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