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Metamath Proof Explorer

Theorem mddmd

Description: The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion mddmd 
|- ( ( A e. CH /\ B e. CH ) -> ( A MH 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 dmdmd 
 |-  ( ( ( _|_ ` A ) e. CH /\ ( _|_ ` B ) e. CH ) -> ( ( _|_ ` A ) MH* ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` A ) ) MH ( _|_ ` ( _|_ ` B ) ) ) )
4 1 2 3 syl2an 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) MH* ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` A ) ) MH ( _|_ ` ( _|_ ` B ) ) ) )
5 ococ 
 |-  ( A e. CH -> ( _|_ ` ( _|_ ` A ) ) = A )
6 ococ 
 |-  ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B )
7 5 6 breqan12d 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` ( _|_ ` A ) ) MH ( _|_ ` ( _|_ ` B ) ) <-> A MH B ) )
8 4 7 bitr2d 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> ( _|_ ` A ) MH* ( _|_ ` B ) ) )