Theorem qlax5i

Description: One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	qlax.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	qlax.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	qlax5i	$$⊢A{\vee }_{\mathscr{H}}\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		qlax.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		qlax.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chdmj2i	$$⊢\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=A\cap \perp \left(B\right)$$
4	3	oveq2i	$$⊢A{\vee }_{\mathscr{H}}\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=A{\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)$$
5	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
6	1 5	chabs1i	$$⊢A{\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)=A$$
7	4 6	eqtri	$$⊢A{\vee }_{\mathscr{H}}\perp \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)=A$$