Theorem cmcm4i

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	|- A e. CH
	pjoml2.2	|- B e. CH
Assertion	cmcm4i	|- ( A C_H B <-> ( _|_ ` A ) C_H ( _|_ ` B ) )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	|- A e. CH
2		pjoml2.2	|- B e. CH
3	1 2	cmcm2i	|- ( A C_H B <-> A C_H ( _|_ ` B ) )
4	2	choccli	|- ( _|_ ` B ) e. CH
5	1 4	cmcm3i	|- ( A C_H ( _|_ ` B ) <-> ( _|_ ` A ) C_H ( _|_ ` B ) )
6	3 5	bitri	|- ( A C_H B <-> ( _|_ ` A ) C_H ( _|_ ` B ) )