Theorem pjoml5i

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

	Ref	Expression
Hypotheses	pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjoml5i	$$⊢A{\vee }_{\mathscr{H}}\left(\perp \left(A\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\right)=A{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chub1i	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B$$
4	1 2	chjcli	$$⊢A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$
5	1 4	pjoml2i	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B\to A{\vee }_{\mathscr{H}}\left(\perp \left(A\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\right)=A{\vee }_{\mathscr{H}}B$$
6	3 5	ax-mp	$$⊢A{\vee }_{\mathscr{H}}\left(\perp \left(A\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\right)=A{\vee }_{\mathscr{H}}B$$