Metamath Proof Explorer

Theorem shub2

Description: A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion shub2 
|- ( ( A e. SH /\ B e. SH ) -> A C_ ( B vH A ) )

Proof

Step Hyp Ref Expression
1 shub1 
 |-  ( ( A e. SH /\ B e. SH ) -> A C_ ( A vH B ) )
2 shjcom 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( B vH A ) )
3 1 2 sseqtrd 
 |-  ( ( A e. SH /\ B e. SH ) -> A C_ ( B vH A ) )