Theorem mul2lt0rgt0

Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018)

	Ref	Expression
Hypotheses	mul2lt0.1	|- ( ph -> A e. RR )
	mul2lt0.2	|- ( ph -> B e. RR )
	mul2lt0.3	|- ( ph -> ( A x. B ) < 0 )
Assertion	mul2lt0rgt0	|- ( ( ph /\ 0 < B ) -> A < 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	3	adantr	|- ( ( ph /\ 0 < B ) -> ( A x. B ) < 0 )
5	2	adantr	|- ( ( ph /\ 0 < B ) -> B e. RR )
6	5	recnd	|- ( ( ph /\ 0 < B ) -> B e. CC )
7	6	mul02d	|- ( ( ph /\ 0 < B ) -> ( 0 x. B ) = 0 )
8	4 7	breqtrrd	|- ( ( ph /\ 0 < B ) -> ( A x. B ) < ( 0 x. B ) )
9	1	adantr	|- ( ( ph /\ 0 < B ) -> A e. RR )
10		0red	|- ( ( ph /\ 0 < B ) -> 0 e. RR )
11		simpr	|- ( ( ph /\ 0 < B ) -> 0 < B )
12	5 11	elrpd	|- ( ( ph /\ 0 < B ) -> B e. RR+ )
13	9 10 12	ltmul1d	|- ( ( ph /\ 0 < B ) -> ( A < 0 <-> ( A x. B ) < ( 0 x. B ) ) )
14	8 13	mpbird	|- ( ( ph /\ 0 < B ) -> A < 0 )