Theorem pjoml5

Description: The orthomodular law. Remark in Kalmbach p. 22. (Contributed by NM, 12-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjoml5	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(\perp \left(A\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\right)=A{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$
3		chub1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\subseteq A{\vee }_{\mathscr{H}}B$$
4		pjoml2	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}\wedge A\subseteq A{\vee }_{\mathscr{H}}B\right)\to A{\vee }_{\mathscr{H}}\left(\perp \left(A\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\right)=A{\vee }_{\mathscr{H}}B$$
5	1 2 3 4	syl3anc	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(\perp \left(A\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\right)=A{\vee }_{\mathscr{H}}B$$