Theorem mul2lt0lgt0

Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 2-Oct-2018)

	Ref	Expression
Hypotheses	mul2lt0.1	|- ( ph -> A e. RR )
	mul2lt0.2	|- ( ph -> B e. RR )
	mul2lt0.3	|- ( ph -> ( A x. B ) < 0 )
Assertion	mul2lt0lgt0	|- ( ( ph /\ 0 < A ) -> B < 0 )

Proof

Step	Hyp	Ref	Expression
1		mul2lt0.1	|- ( ph -> A e. RR )
2		mul2lt0.2	|- ( ph -> B e. RR )
3		mul2lt0.3	|- ( ph -> ( A x. B ) < 0 )
4	1	recnd	|- ( ph -> A e. CC )
5	2	recnd	|- ( ph -> B e. CC )
6	4 5	mulcomd	|- ( ph -> ( A x. B ) = ( B x. A ) )
7	6 3	eqbrtrrd	|- ( ph -> ( B x. A ) < 0 )
8	2 1 7	mul2lt0rgt0	|- ( ( ph /\ 0 < A ) -> B < 0 )