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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