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	|- ( ( A e. CH /\ B e. CH ) -> ( A vH ( ( _|_ ` A ) i^i ( A vH B ) ) ) = ( A vH B ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( A e. CH /\ B e. CH ) -> A e. CH )
2		chjcl	|- ( ( A e. CH /\ B e. CH ) -> ( A vH B ) e. CH )
3		chub1	|- ( ( A e. CH /\ B e. CH ) -> A C_ ( A vH B ) )
4		pjoml2	|- ( ( A e. CH /\ ( A vH B ) e. CH /\ A C_ ( A vH B ) ) -> ( A vH ( ( _|_ ` A ) i^i ( A vH B ) ) ) = ( A vH B ) )
5	1 2 3 4	syl3anc	|- ( ( A e. CH /\ B e. CH ) -> ( A vH ( ( _|_ ` A ) i^i ( A vH B ) ) ) = ( A vH B ) )