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Metamath Proof Explorer

Theorem pmapval

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)

Ref Expression
Hypotheses pmapfval.b 
|- B = ( Base ` K )
pmapfval.l 
|- .<_ = ( le ` K )
pmapfval.a 
|- A = ( Atoms ` K )
pmapfval.m 
|- M = ( pmap ` K )
Assertion pmapval 
|- ( ( K e. C /\ X e. B ) -> ( M ` X ) = { a e. A | a .<_ X } )

Proof

Step Hyp Ref Expression
1 pmapfval.b 
 |-  B = ( Base ` K )
2 pmapfval.l 
 |-  .<_ = ( le ` K )
3 pmapfval.a 
 |-  A = ( Atoms ` K )
4 pmapfval.m 
 |-  M = ( pmap ` K )
5 1 2 3 4 pmapfval 
 |-  ( K e. C -> M = ( x e. B |-> { a e. A | a .<_ x } ) )
6 5 fveq1d 
 |-  ( K e. C -> ( M ` X ) = ( ( x e. B |-> { a e. A | a .<_ x } ) ` X ) )
7 breq2 
 |-  ( x = X -> ( a .<_ x <-> a .<_ X ) )
8 7 rabbidv 
 |-  ( x = X -> { a e. A | a .<_ x } = { a e. A | a .<_ X } )
9 eqid 
 |-  ( x e. B |-> { a e. A | a .<_ x } ) = ( x e. B |-> { a e. A | a .<_ x } )
10 3 fvexi 
 |-  A e. _V
11 10 rabex 
 |-  { a e. A | a .<_ X } e. _V
12 8 9 11 fvmpt 
 |-  ( X e. B -> ( ( x e. B |-> { a e. A | a .<_ x } ) ` X ) = { a e. A | a .<_ X } )
13 6 12 sylan9eq 
 |-  ( ( K e. C /\ X e. B ) -> ( M ` X ) = { a e. A | a .<_ X } )