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Metamath Proof Explorer

Theorem mulltgt0d

Description: Negative times positive is negative. (Contributed by SN, 26-Nov-2025)

Ref Expression
Hypotheses mullt0b1d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
mullt0b1d.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
mullt0b1d.1 ⊢ ( 𝜑 → 𝐴 < 0 )
mulltgt0d.2 ⊢ ( 𝜑 → 0 < 𝐵 )
Assertion mulltgt0d ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 )

Proof

Step Hyp Ref Expression
1 mullt0b1d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 mullt0b1d.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 mullt0b1d.1 ⊢ ( 𝜑 → 𝐴 < 0 )
4 mulltgt0d.2 ⊢ ( 𝜑 → 0 < 𝐵 )
5 3 lt0ne0d ⊢ ( 𝜑 → 𝐴 ≠ 0 )
6 4 gt0ne0d ⊢ ( 𝜑 → 𝐵 ≠ 0 )
7 5 6 jca ⊢ ( 𝜑 → ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) )
8 neanior ⊢ ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) )
9 7 8 sylib ⊢ ( 𝜑 → ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) )
10 1 2 sn-remul0ord ⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) )
11 9 10 mtbird ⊢ ( 𝜑 → ¬ ( 𝐴 · 𝐵 ) = 0 )
12 0red ⊢ ( 𝜑 → 0 ∈ ℝ )
13 1 12 3 ltnsymd ⊢ ( 𝜑 → ¬ 0 < 𝐴 )
14 1 2 4 mulgt0b2d ⊢ ( 𝜑 → ( 0 < 𝐴 ↔ 0 < ( 𝐴 · 𝐵 ) ) )
15 13 14 mtbid ⊢ ( 𝜑 → ¬ 0 < ( 𝐴 · 𝐵 ) )
16 ioran ⊢ ( ¬ ( ( 𝐴 · 𝐵 ) = 0 ∨ 0 < ( 𝐴 · 𝐵 ) ) ↔ ( ¬ ( 𝐴 · 𝐵 ) = 0 ∧ ¬ 0 < ( 𝐴 · 𝐵 ) ) )
17 11 15 16 sylanbrc ⊢ ( 𝜑 → ¬ ( ( 𝐴 · 𝐵 ) = 0 ∨ 0 < ( 𝐴 · 𝐵 ) ) )
18 1 2 remulcld ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℝ )
19 18 12 lttrid ⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) < 0 ↔ ¬ ( ( 𝐴 · 𝐵 ) = 0 ∨ 0 < ( 𝐴 · 𝐵 ) ) ) )
20 17 19 mpbird ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) < 0 )