Metamath Proof Explorer

Theorem chdmj2

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

Ref Expression
Assertion chdmj2 
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) vH B ) ) = ( A i^i ( _|_ ` B ) ) )

Proof

Step Hyp Ref Expression
1 choccl 
 |-  ( A e. CH -> ( _|_ ` A ) e. CH )
2 chdmj1 
 |-  ( ( ( _|_ ` A ) e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) vH B ) ) = ( ( _|_ ` ( _|_ ` A ) ) i^i ( _|_ ` B ) ) )
3 1 2 sylan 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) vH B ) ) = ( ( _|_ ` ( _|_ ` A ) ) i^i ( _|_ ` B ) ) )
4 ococ 
 |-  ( A e. CH -> ( _|_ ` ( _|_ ` A ) ) = A )
5 4 ineq1d 
 |-  ( A e. CH -> ( ( _|_ ` ( _|_ ` A ) ) i^i ( _|_ ` B ) ) = ( A i^i ( _|_ ` B ) ) )
6 5 adantr 
 |-  ( ( 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 ) ) )