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Theorem pmapfval 29945
Description: The projective map of a Hilbert lattice. (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
pmapfval  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
Distinct variable groups:    A, a    x, B    x, a, K
Allowed substitution hints:    A( x)    B( a)    C( x, a)    .<_ ( x, a)    M( x, a)

Proof of Theorem pmapfval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 pmapfval.m . . 3  |-  M  =  ( pmap `  K
)
3 fveq2 5525 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 pmapfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
7 pmapfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
9 fveq2 5525 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
10 pmapfval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1211breqd 4034 . . . . . 6  |-  ( k  =  K  ->  (
a ( le `  k ) x  <->  a  .<_  x ) )
138, 12rabeqbidv 2783 . . . . 5  |-  ( k  =  K  ->  { a  e.  ( Atoms `  k
)  |  a ( le `  k ) x }  =  {
a  e.  A  | 
a  .<_  x } )
145, 13mpteq12dv 4098 . . . 4  |-  ( k  =  K  ->  (
x  e.  ( Base `  k )  |->  { a  e.  ( Atoms `  k
)  |  a ( le `  k ) x } )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) )
15 df-pmap 29693 . . . 4  |-  pmap  =  ( k  e.  _V  |->  ( x  e.  ( Base `  k )  |->  { a  e.  ( Atoms `  k )  |  a ( le `  k
) x } ) )
16 fvex 5539 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2353 . . . . 5  |-  B  e. 
_V
1817mptex 5746 . . . 4  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  e. 
_V
1914, 15, 18fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( pmap `  K )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) )
202, 19syl5eq 2327 . 2  |-  ( K  e.  _V  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
211, 20syl 15 1  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  pmapval  29946
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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