Theorem cmcm2

Description: Commutation with orthocomplement. Theorem 2.3(i) of Beran p. 39. (Contributed by NM, 14-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	cmcm2	|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A C_H ( _|_ ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		cmcm3	|- ( ( B e. CH /\ A e. CH ) -> ( B C_H A <-> ( _|_ ` B ) C_H A ) )
2	1	ancoms	|- ( ( A e. CH /\ B e. CH ) -> ( B C_H A <-> ( _|_ ` B ) C_H A ) )
3		cmcm	|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> B C_H A ) )
4		choccl	|- ( B e. CH -> ( _|_ ` B ) e. CH )
5		cmcm	|- ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( A C_H ( _|_ ` B ) <-> ( _|_ ` B ) C_H A ) )
6	4 5	sylan2	|- ( ( A e. CH /\ B e. CH ) -> ( A C_H ( _|_ ` B ) <-> ( _|_ ` B ) C_H A ) )
7	2 3 6	3bitr4d	|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A C_H ( _|_ ` B ) ) )