Theorem chdmj2i

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

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chdmj2i	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=A\cap \perp \left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4	3 2	chdmj1i	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=\perp \left(\perp \left(A\right)\right)\cap \perp \left(B\right)$$
5	1	pjococi	$$⊢\perp \left(\perp \left(A\right)\right)=A$$
6	5	ineq1i	$$⊢\perp \left(\perp \left(A\right)\right)\cap \perp \left(B\right)=A\cap \perp \left(B\right)$$
7	4 6	eqtri	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=A\cap \perp \left(B\right)$$