Theorem qlax4i

Description: One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-4" 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	qlax4i	$$⊢A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}\perp \left(B\right)\right)=B{\vee }_{\mathscr{H}}\perp \left(B\right)$$

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	chj1i	$$⊢A{\vee }_{\mathscr{H}}\mathscr{H}=\mathscr{H}$$
4	2	chjoi	$$⊢B{\vee }_{\mathscr{H}}\perp \left(B\right)=\mathscr{H}$$
5	4	oveq2i	$$⊢A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}\perp \left(B\right)\right)=A{\vee }_{\mathscr{H}}\mathscr{H}$$
6	3 5 4	3eqtr4i	$$⊢A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}\perp \left(B\right)\right)=B{\vee }_{\mathscr{H}}\perp \left(B\right)$$