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Metamath Proof Explorer

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 )