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

Metamath Proof Explorer

Theorem prodgt02

Description: Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005)

Ref Expression
Assertion prodgt02 ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐵 ∧ 0 < ( 𝐴 · 𝐵 ) ) ) → 0 < 𝐴 )

Proof

Step Hyp Ref Expression
1 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2 recn ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
3 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
5 4 breq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < ( 𝐴 · 𝐵 ) ↔ 0 < ( 𝐵 · 𝐴 ) ) )
6 5 biimpd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < ( 𝐴 · 𝐵 ) → 0 < ( 𝐵 · 𝐴 ) ) )
7 prodgt0 ⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ ( 0 ≤ 𝐵 ∧ 0 < ( 𝐵 · 𝐴 ) ) ) → 0 < 𝐴 )
8 7 ex ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 0 ≤ 𝐵 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 0 < 𝐴 ) )
9 8 ancoms ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 ≤ 𝐵 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 0 < 𝐴 ) )
10 6 9 sylan2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 ≤ 𝐵 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 < 𝐴 ) )
11 10 imp ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐵 ∧ 0 < ( 𝐴 · 𝐵 ) ) ) → 0 < 𝐴 )