Theorem cvdmd

Description: The covering property implies the dual modular pair property. Lemma 7.5.2 of MaedaMaeda p. 31. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvdmd	|- ( ( A e. CH /\ B e. CH /\ B A MH* B )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( A e. CH -> ( _|_ ` A ) e. CH )
2		choccl	|- ( B e. CH -> ( _|_ ` B ) e. CH )
3		cvmd	|- ( ( ( _|_ ` A ) e. CH /\ ( _|_ ` B ) e. CH /\ ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ( _|_ ` A ) MH ( _|_ ` B ) )
4	3	3expia	|- ( ( ( _|_ ` A ) e. CH /\ ( _|_ ` B ) e. CH ) -> ( ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ( _|_ ` A ) MH ( _|_ ` B ) ) )
5	1 2 4	syl2an	|- ( ( A e. CH /\ B e. CH ) -> ( ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ( _|_ ` A ) MH ( _|_ ` B ) ) )
6		simpr	|- ( ( A e. CH /\ B e. CH ) -> B e. CH )
7		chjcl	|- ( ( A e. CH /\ B e. CH ) -> ( A vH B ) e. CH )
8		cvcon3	|- ( ( B e. CH /\ ( A vH B ) e. CH ) -> ( B ( _|_ ` ( A vH B ) )
9	6 7 8	syl2anc	|- ( ( A e. CH /\ B e. CH ) -> ( B ( _|_ ` ( A vH B ) )
10		chdmj1	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A vH B ) ) = ( ( _|_ ` A ) i^i ( _|_ ` B ) ) )
11	10	breq1d	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` ( A vH B ) ) ( ( _|_ ` A ) i^i ( _|_ ` B ) )
12	9 11	bitrd	|- ( ( A e. CH /\ B e. CH ) -> ( B ( ( _|_ ` A ) i^i ( _|_ ` B ) )
13		dmdmd	|- ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> ( _|_ ` A ) MH ( _|_ ` B ) ) )
14	5 12 13	3imtr4d	|- ( ( A e. CH /\ B e. CH ) -> ( B A MH* B ) )
15	14	3impia	|- ( ( A e. CH /\ B e. CH /\ B A MH* B )