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	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	mulgt0con1dlem.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	mulgt0con1dlem.1	⊢ ( 𝜑 → ( 0 < 𝐴 → 0 < 𝐵 ) )
	mulgt0con1dlem.2	⊢ ( 𝜑 → ( 𝐴 = 0 → 𝐵 = 0 ) )
Assertion	mulgt0con1dlem	⊢ ( 𝜑 → ( 𝐵 < 0 → 𝐴 < 0 ) )

Proof

Step	Hyp	Ref	Expression
1		mulgt0con1dlem.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		mulgt0con1dlem.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		mulgt0con1dlem.1	⊢ ( 𝜑 → ( 0 < 𝐴 → 0 < 𝐵 ) )
4		mulgt0con1dlem.2	⊢ ( 𝜑 → ( 𝐴 = 0 → 𝐵 = 0 ) )
5		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
6	2 5	lttrid	⊢ ( 𝜑 → ( 𝐵 < 0 ↔ ¬ ( 𝐵 = 0 ∨ 0 < 𝐵 ) ) )
7	4 3	orim12d	⊢ ( 𝜑 → ( ( 𝐴 = 0 ∨ 0 < 𝐴 ) → ( 𝐵 = 0 ∨ 0 < 𝐵 ) ) )
8	7	con3d	⊢ ( 𝜑 → ( ¬ ( 𝐵 = 0 ∨ 0 < 𝐵 ) → ¬ ( 𝐴 = 0 ∨ 0 < 𝐴 ) ) )
9	1 5	lttrid	⊢ ( 𝜑 → ( 𝐴 < 0 ↔ ¬ ( 𝐴 = 0 ∨ 0 < 𝐴 ) ) )
10	8 9	sylibrd	⊢ ( 𝜑 → ( ¬ ( 𝐵 = 0 ∨ 0 < 𝐵 ) → 𝐴 < 0 ) )
11	6 10	sylbid	⊢ ( 𝜑 → ( 𝐵 < 0 → 𝐴 < 0 ) )