Metamath Proof Explorer

Theorem shjval

Description: Value of join in SH . (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

Ref Expression
Assertion shjval 
|- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )

Proof

Step Hyp Ref Expression
1 shss 
 |-  ( A e. SH -> A C_ ~H )
2 shss 
 |-  ( B e. SH -> B C_ ~H )
3 sshjval 
 |-  ( ( A C_ ~H /\ B C_ ~H ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
4 1 2 3 syl2an 
 |-  ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )