Metamath Proof Explorer

Theorem shincl

Description: Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 ineq1 
 |-  ( A = if ( A e. SH , A , ~H ) -> ( A i^i B ) = ( if ( A e. SH , A , ~H ) i^i B ) )
2 1 eleq1d 
 |-  ( A = if ( A e. SH , A , ~H ) -> ( ( A i^i B ) e. SH <-> ( if ( A e. SH , A , ~H ) i^i B ) e. SH ) )
3 ineq2 
 |-  ( B = if ( B e. SH , B , ~H ) -> ( if ( A e. SH , A , ~H ) i^i B ) = ( if ( A e. SH , A , ~H ) i^i if ( B e. SH , B , ~H ) ) )
4 3 eleq1d 
 |-  ( B = if ( B e. SH , B , ~H ) -> ( ( if ( A e. SH , A , ~H ) i^i B ) e. SH <-> ( if ( A e. SH , A , ~H ) i^i if ( B e. SH , B , ~H ) ) e. SH ) )
5 helsh 
 |-  ~H e. SH
6 5 elimel 
 |-  if ( A e. SH , A , ~H ) e. SH
7 5 elimel 
 |-  if ( B e. SH , B , ~H ) e. SH
8 6 7 shincli 
 |-  ( if ( A e. SH , A , ~H ) i^i if ( B e. SH , B , ~H ) ) e. SH
9 2 4 8 dedth2h 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A i^i B ) e. SH )