Metamath Proof Explorer

Theorem dmdi2

Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005) (New usage is discouraged.)

		Ref	Expression
	Assertion	dmdi2	\|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH* B /\ B C_ C ) ) -> ( C i^i ( A vH B ) ) C_ ( ( C i^i A ) vH B ) )

Step	Hyp	Ref	Expression
1		dmdi	\|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH* B /\ B C_ C ) ) -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) )
2		eqimss2	\|- ( ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) -> ( C i^i ( A vH B ) ) C_ ( ( C i^i A ) vH B ) )
3	1 2	syl	\|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH* B /\ B C_ C ) ) -> ( C i^i ( A vH B ) ) C_ ( ( C i^i A ) vH B ) )

Step

Hyp

Ref

Expression

 |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH* B /\ B C_ C ) ) -> ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) )

 |-  ( ( ( C i^i A ) vH B ) = ( C i^i ( A vH B ) ) -> ( C i^i ( A vH B ) ) C_ ( ( C i^i A ) vH B ) )

1 2

 |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH* B /\ B C_ C ) ) -> ( C i^i ( A vH B ) ) C_ ( ( C i^i A ) vH B ) )