Theorem addneg1mul

Description: Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021)

	Ref	Expression
Assertion	addneg1mul	|- ( ( A e. CC /\ B e. CC ) -> ( A + ( -u 1 x. B ) ) = ( A - B ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1	|- ( B e. CC -> ( -u 1 x. B ) = -u B )
2	1	adantl	|- ( ( A e. CC /\ B e. CC ) -> ( -u 1 x. B ) = -u B )
3	2	oveq2d	|- ( ( A e. CC /\ B e. CC ) -> ( A + ( -u 1 x. B ) ) = ( A + -u B ) )
4		negsub	|- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
5	3 4	eqtrd	|- ( ( A e. CC /\ B e. CC ) -> ( A + ( -u 1 x. B ) ) = ( A - B ) )