Theorem ssdmd1

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	ssdmd1	|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> A MH* B )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( B e. CH -> ( _|_ ` B ) e. CH )
2		choccl	|- ( A e. CH -> ( _|_ ` A ) e. CH )
3		ssmd2	|- ( ( ( _|_ ` B ) e. CH /\ ( _|_ ` A ) e. CH /\ ( _|_ ` B ) C_ ( _|_ ` A ) ) -> ( _|_ ` A ) MH ( _|_ ` B ) )
4	3	3expia	|- ( ( ( _|_ ` B ) e. CH /\ ( _|_ ` A ) e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) -> ( _|_ ` A ) MH ( _|_ ` B ) ) )
5	1 2 4	syl2anr	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) -> ( _|_ ` A ) MH ( _|_ ` B ) ) )
6		chsscon3	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) )
7		dmdmd	|- ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> ( _|_ ` A ) MH ( _|_ ` B ) ) )
8	5 6 7	3imtr4d	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B -> A MH* B ) )
9	8	3impia	|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> A MH* B )