Theorem mulgt1d

Description: The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	divgt0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	mulgt1d.3	⊢ ( 𝜑 → 1 < 𝐴 )
	mulgt1d.4	⊢ ( 𝜑 → 1 < 𝐵 )
Assertion	mulgt1d	⊢ ( 𝜑 → 1 < ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		divgt0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		mulgt1d.3	⊢ ( 𝜑 → 1 < 𝐴 )
4		mulgt1d.4	⊢ ( 𝜑 → 1 < 𝐵 )
5		mulgt1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 1 < 𝐴 ∧ 1 < 𝐵 ) ) → 1 < ( 𝐴 · 𝐵 ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → 1 < ( 𝐴 · 𝐵 ) )