Theorem ltmulgt12d

Description: Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	ltmulgt12d	⊢ ( 𝜑 → ( 1 < 𝐴 ↔ 𝐵 < ( 𝐴 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3	2	rpred	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4	2	rpgt0d	⊢ ( 𝜑 → 0 < 𝐵 )
5		ltmulgt12	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 0 < 𝐵 ) → ( 1 < 𝐴 ↔ 𝐵 < ( 𝐴 · 𝐵 ) ) )
6	3 1 4 5	syl3anc	⊢ ( 𝜑 → ( 1 < 𝐴 ↔ 𝐵 < ( 𝐴 · 𝐵 ) ) )