Metamath Proof Explorer

Theorem honegdi

Description: Distribution of negative over addition. (Contributed by NM, 24-Aug-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	honegdi	\|- ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( -u 1 .op ( T +op U ) ) = ( ( -u 1 .op T ) +op ( -u 1 .op U ) ) )

Step	Hyp	Ref	Expression
1		neg1cn	\|- -u 1 e. CC
2		hoadddi	\|- ( ( -u 1 e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( -u 1 .op ( T +op U ) ) = ( ( -u 1 .op T ) +op ( -u 1 .op U ) ) )
3	1 2	mp3an1	\|- ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( -u 1 .op ( T +op U ) ) = ( ( -u 1 .op T ) +op ( -u 1 .op U ) ) )

Step

Hyp

Ref

Expression

 |-  -u 1 e. CC

 |-  ( ( -u 1 e. CC /\ T : ~H --> ~H /\ U : ~H --> ~H ) -> ( -u 1 .op ( T +op U ) ) = ( ( -u 1 .op T ) +op ( -u 1 .op U ) ) )

1 2

 |-  ( ( T : ~H --> ~H /\ U : ~H --> ~H ) -> ( -u 1 .op ( T +op U ) ) = ( ( -u 1 .op T ) +op ( -u 1 .op U ) ) )