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Theorem pmapfval 30567
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 2809 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 pmapfval.m . . 3  |-  M  =  ( pmap `  K
)
3 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 pmapfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
7 pmapfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
9 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
10 pmapfval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1211breqd 4050 . . . . . 6  |-  ( k  =  K  ->  (
a ( le `  k ) x  <->  a  .<_  x ) )
138, 12rabeqbidv 2796 . . . . 5  |-  ( k  =  K  ->  { a  e.  ( Atoms `  k
)  |  a ( le `  k ) x }  =  {
a  e.  A  | 
a  .<_  x } )
145, 13mpteq12dv 4114 . . . 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 30315 . . . 4  |-  pmap  =  ( k  e.  _V  |->  ( x  e.  ( Base `  k )  |->  { a  e.  ( Atoms `  k )  |  a ( le `  k
) x } ) )
16 fvex 5555 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2366 . . . . 5  |-  B  e. 
_V
1817mptex 5762 . . . 4  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  e. 
_V
1914, 15, 18fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( pmap `  K )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) )
202, 19syl5eq 2340 . 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 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:  pmapval  30568
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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