Theorem chdmj4i

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	chdmj4i	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}\perp \left(B\right)\right)=A\cap B$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4	1 3	chdmj2i	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}\perp \left(B\right)\right)=A\cap \perp \left(\perp \left(B\right)\right)$$
5	2	pjococi	$$⊢\perp \left(\perp \left(B\right)\right)=B$$
6	5	ineq2i	$$⊢A\cap \perp \left(\perp \left(B\right)\right)=A\cap B$$
7	4 6	eqtri	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}\perp \left(B\right)\right)=A\cap B$$