Theorem mulgt0con1dlem

Description: Lemma for mulgt0con1d . Contraposes a positive deduction to a negative deduction. (Contributed by SN, 26-Jun-2024)

	Ref	Expression
Hypotheses	mulgt0con1dlem.a	$$⊢\phi \to A\in ℝ$$
	mulgt0con1dlem.b	$$⊢\phi \to B\in ℝ$$
	mulgt0con1dlem.1	$$⊢\phi \to \left(0<A\to 0<B\right)$$
	mulgt0con1dlem.2	$$⊢\phi \to \left(A=0\to B=0\right)$$
Assertion	mulgt0con1dlem	$$⊢\phi \to \left(B<0\to A<0\right)$$

Proof

Step	Hyp	Ref	Expression
1		mulgt0con1dlem.a	$$⊢\phi \to A\in ℝ$$
2		mulgt0con1dlem.b	$$⊢\phi \to B\in ℝ$$
3		mulgt0con1dlem.1	$$⊢\phi \to \left(0<A\to 0<B\right)$$
4		mulgt0con1dlem.2	$$⊢\phi \to \left(A=0\to B=0\right)$$
5		0red	$$⊢\phi \to 0\in ℝ$$
6	2 5	lttrid	$$⊢\phi \to \left(B<0\leftrightarrow \neg \left(B=0\vee 0<B\right)\right)$$
7	4 3	orim12d	$$⊢\phi \to \left(\left(A=0\vee 0<A\right)\to \left(B=0\vee 0<B\right)\right)$$
8	7	con3d	$$⊢\phi \to \left(\neg \left(B=0\vee 0<B\right)\to \neg \left(A=0\vee 0<A\right)\right)$$
9	1 5	lttrid	$$⊢\phi \to \left(A<0\leftrightarrow \neg \left(A=0\vee 0<A\right)\right)$$
10	8 9	sylibrd	$$⊢\phi \to \left(\neg \left(B=0\vee 0<B\right)\to A<0\right)$$
11	6 10	sylbid	$$⊢\phi \to \left(B<0\to A<0\right)$$