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Metamath Proof Explorer

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 )