Theorem cmcm3i

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 4-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	|- A e. CH
	pjoml2.2	|- B e. CH
Assertion	cmcm3i	|- ( 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	2 1	cmcm2i	|- ( B C_H A <-> B C_H ( _|_ ` A ) )
4	1 2	cmcmi	|- ( A C_H B <-> B C_H A )
5	1	choccli	|- ( _|_ ` A ) e. CH
6	5 2	cmcmi	|- ( ( _|_ ` A ) C_H B <-> B C_H ( _|_ ` A ) )
7	3 4 6	3bitr4i	|- ( A C_H B <-> ( _|_ ` A ) C_H B )