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Theorem pmapfval 33398
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 2980 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 pmapfval.m . . 3  |-  M  =  ( pmap `  K
)
3 fveq2 5690 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 pmapfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2492 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5690 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
7 pmapfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7syl6eqr 2492 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
9 fveq2 5690 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
10 pmapfval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
119, 10syl6eqr 2492 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1211breqd 4302 . . . . . 6  |-  ( k  =  K  ->  (
a ( le `  k ) x  <->  a  .<_  x ) )
138, 12rabeqbidv 2966 . . . . 5  |-  ( k  =  K  ->  { a  e.  ( Atoms `  k
)  |  a ( le `  k ) x }  =  {
a  e.  A  | 
a  .<_  x } )
145, 13mpteq12dv 4369 . . . 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 33146 . . . 4  |-  pmap  =  ( k  e.  _V  |->  ( x  e.  ( Base `  k )  |->  { a  e.  ( Atoms `  k )  |  a ( le `  k
) x } ) )
16 fvex 5700 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2512 . . . . 5  |-  B  e. 
_V
1817mptex 5947 . . . 4  |-  ( x  e.  B  |->  { a  e.  A  |  a 
.<_  x } )  e. 
_V
1914, 15, 18fvmpt 5773 . . 3  |-  ( K  e.  _V  ->  ( pmap `  K )  =  ( x  e.  B  |->  { a  e.  A  |  a  .<_  x }
) )
202, 19syl5eq 2486 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2718   _Vcvv 2971   class class class wbr 4291    e. cmpt 4349   ` cfv 5417   Basecbs 14173   lecple 14244   Atomscatm 32906   pmapcpmap 33139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-pmap 33146
This theorem is referenced by:  pmapval  33399
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