Metamath Proof Explorer

Theorem chincl

Description: Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chincl 
|- ( ( A e. CH /\ B e. CH ) -> ( A i^i B ) e. CH )

Proof

Step Hyp Ref Expression
1 ineq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A i^i B ) = ( if ( A e. CH , A , ~H ) i^i B ) )
2 1 eleq1d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A i^i B ) e. CH <-> ( if ( A e. CH , A , ~H ) i^i B ) e. CH ) )
3 ineq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) i^i B ) = ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) )
4 3 eleq1d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) i^i B ) e. CH <-> ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) e. CH ) )
5 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
6 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
7 5 6 chincli 
 |-  ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) e. CH
8 2 4 7 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A i^i B ) e. CH )