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Theorem pmapfval 30553
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 2964 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 pmapfval.m . . 3  |-  M  =  ( pmap `  K
)
3 fveq2 5728 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 pmapfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2486 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5728 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
7 pmapfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7syl6eqr 2486 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
9 fveq2 5728 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
10 pmapfval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
119, 10syl6eqr 2486 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1211breqd 4223 . . . . . 6  |-  ( k  =  K  ->  (
a ( le `  k ) x  <->  a  .<_  x ) )
138, 12rabeqbidv 2951 . . . . 5  |-  ( k  =  K  ->  { a  e.  ( Atoms `  k
)  |  a ( le `  k ) x }  =  {
a  e.  A  | 
a  .<_  x } )
145, 13mpteq12dv 4287 . . . 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 30301 . . . 4  |-  pmap  =  ( k  e.  _V  |->  ( x  e.  ( Base `  k )  |->  { a  e.  ( Atoms `  k )  |  a ( le `  k
) x } ) )
16 fvex 5742 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2506 . . . . 5  |-  B  e. 
_V
1817mptex 5966 . . . 4  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  e. 
_V
1914, 15, 18fvmpt 5806 . . 3  |-  ( K  e.  _V  ->  ( pmap `  K )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) )
202, 19syl5eq 2480 . 2  |-  ( K  e.  _V  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
211, 20syl 16 1  |-  ( K  e.  C  ->  M  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ` cfv 5454   Basecbs 13469   lecple 13536   Atomscatm 30061   pmapcpmap 30294
This theorem is referenced by:  pmapval  30554
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-pmap 30301
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