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 )