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Theorem pmapval 29946
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 29945 . . 3  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
65fveq1d 5527 . 2  |-  ( K  e.  C  ->  ( M `  X )  =  ( ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } ) `  X ) )
7 breq2 4027 . . . 4  |-  ( x  =  X  ->  (
a  .<_  x  <->  a  .<_  X ) )
87rabbidv 2780 . . 3  |-  ( x  =  X  ->  { a  e.  A  |  a 
.<_  x }  =  {
a  e.  A  | 
a  .<_  X } )
9 eqid 2283 . . 3  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
)
10 fvex 5539 . . . . 5  |-  ( Atoms `  K )  e.  _V
113, 10eqeltri 2353 . . . 4  |-  A  e. 
_V
1211rabex 4165 . . 3  |-  { a  e.  A  |  a 
.<_  X }  e.  _V
138, 9, 12fvmpt 5602 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) `  X )  =  { a  e.  A  |  a  .<_  X }
)
146, 13sylan9eq 2335 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255   Basecbs 13148   lecple 13215   Atomscatm 29453   pmapcpmap 29686
This theorem is referenced by:  elpmap  29947  pmapssat  29948  pmaple  29950  pmapat  29952  pmap0  29954  pmap1N  29956  pmapsub  29957  pmapglbx  29958  isline2  29963  linepmap  29964  polpmapN  30101  2polssN  30104  pmaplubN  30113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-pmap 29693
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