Theorem chdmm3

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

	Ref	Expression
Assertion	chdmm3	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A i^i ( _|_ ` B ) ) ) = ( ( _|_ ` A ) vH B ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( B e. CH -> ( _|_ ` B ) e. CH )
2		chdmm1	|- ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( _|_ ` ( A i^i ( _|_ ` B ) ) ) = ( ( _|_ ` A ) vH ( _|_ ` ( _|_ ` B ) ) ) )
3	1 2	sylan2	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A i^i ( _|_ ` B ) ) ) = ( ( _|_ ` A ) vH ( _|_ ` ( _|_ ` B ) ) ) )
4		ococ	|- ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B )
5	4	adantl	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` B ) ) = B )
6	5	oveq2d	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) vH ( _|_ ` ( _|_ ` B ) ) ) = ( ( _|_ ` A ) vH B ) )
7	3 6	eqtrd	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A i^i ( _|_ ` B ) ) ) = ( ( _|_ ` A ) vH B ) )