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Metamath Proof Explorer

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	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
		mul2lt0.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
		mul2lt0.3	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 )
	Assertion	mul2lt0rgt0	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐴 < 0 )

Proof

Step	Hyp	Ref	Expression
1		mul2lt0.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		mul2lt0.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		mul2lt0.3	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 )
4	3	adantr	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → ( 𝐴 · 𝐵 ) < 0 )
5	2	adantr	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐵 ∈ ℝ )
6	5	recnd	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐵 ∈ ℂ )
7	6	mul02d	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → ( 0 · 𝐵 ) = 0 )
8	4 7	breqtrrd	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → ( 𝐴 · 𝐵 ) < ( 0 · 𝐵 ) )
9	1	adantr	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐴 ∈ ℝ )
10		0red	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 0 ∈ ℝ )
11		simpr	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 0 < 𝐵 )
12	5 11	elrpd	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐵 ∈ ℝ⁺ )
13	9 10 12	ltmul1d	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → ( 𝐴 < 0 ↔ ( 𝐴 · 𝐵 ) < ( 0 · 𝐵 ) ) )
14	8 13	mpbird	⊢ ( ( 𝜑 ∧ 0 < 𝐵 ) → 𝐴 < 0 )