This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )