Theorem ssdmd2

Description: Ordering implies the dual modular pair property. Remark in MaedaMaeda p. 1. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	ssdmd2	|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> ( _|_ ` B ) MH ( _|_ ` A ) )

Proof

Step	Hyp	Ref	Expression
1		chsscon3	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) )
2		choccl	|- ( B e. CH -> ( _|_ ` B ) e. CH )
3		choccl	|- ( A e. CH -> ( _|_ ` A ) e. CH )
4		ssmd1	|- ( ( ( _|_ ` B ) e. CH /\ ( _|_ ` A ) e. CH /\ ( _|_ ` B ) C_ ( _|_ ` A ) ) -> ( _|_ ` B ) MH ( _|_ ` A ) )
5	4	3expia	|- ( ( ( _|_ ` B ) e. CH /\ ( _|_ ` A ) e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) -> ( _|_ ` B ) MH ( _|_ ` A ) ) )
6	2 3 5	syl2anr	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) -> ( _|_ ` B ) MH ( _|_ ` A ) ) )
7	1 6	sylbid	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B -> ( _|_ ` B ) MH ( _|_ ` A ) ) )
8	7	3impia	|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> ( _|_ ` B ) MH ( _|_ ` A ) )