Metamath Proof Explorer

Theorem chdmj3

Description: De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chdmj3 
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A vH ( _|_ ` B ) ) ) = ( ( _|_ ` A ) i^i B ) )

Proof

Step Hyp Ref Expression
1 choccl 
 |-  ( B e. CH -> ( _|_ ` B ) e. CH )
2 chdmj1 
 |-  ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( _|_ ` ( A vH ( _|_ ` B ) ) ) = ( ( _|_ ` A ) i^i ( _|_ ` ( _|_ ` B ) ) ) )
3 1 2 sylan2 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A vH ( _|_ ` B ) ) ) = ( ( _|_ ` A ) i^i ( _|_ ` ( _|_ ` B ) ) ) )
4 ococ 
 |-  ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B )
5 4 adantl 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` B ) ) = B )
6 5 ineq2d 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) i^i ( _|_ ` ( _|_ ` B ) ) ) = ( ( _|_ ` A ) i^i B ) )
7 3 6 eqtrd 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A vH ( _|_ ` B ) ) ) = ( ( _|_ ` A ) i^i B ) )