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Theorem pmapval 33075
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 33074 . . 3  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
65fveq1d 5874 . 2  |-  ( K  e.  C  ->  ( M `  X )  =  ( ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } ) `  X ) )
7 breq2 4421 . . . 4  |-  ( x  =  X  ->  (
a  .<_  x  <->  a  .<_  X ) )
87rabbidv 3070 . . 3  |-  ( x  =  X  ->  { a  e.  A  |  a 
.<_  x }  =  {
a  e.  A  | 
a  .<_  X } )
9 eqid 2420 . . 3  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
)
10 fvex 5882 . . . . 5  |-  ( Atoms `  K )  e.  _V
113, 10eqeltri 2504 . . . 4  |-  A  e. 
_V
1211rabex 4567 . . 3  |-  { a  e.  A  |  a 
.<_  X }  e.  _V
138, 9, 12fvmpt 5955 . 2  |-  ( X  e.  B  ->  (
( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) `  X )  =  { a  e.  A  |  a  .<_  X }
)
146, 13sylan9eq 2481 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 370    = wceq 1437    e. wcel 1867   {crab 2777   _Vcvv 3078   class class class wbr 4417    |-> cmpt 4475   ` cfv 5592   Basecbs 15081   lecple 15157   Atomscatm 32582   pmapcpmap 32815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-pmap 32822
This theorem is referenced by:  elpmap  33076  pmapssat  33077  pmaple  33079  pmapat  33081  pmap0  33083  pmap1N  33085  pmapsub  33086  pmapglbx  33087  isline2  33092  linepmap  33093  polpmapN  33230  2polssN  33233  pmaplubN  33242
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