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Metamath Proof Explorer

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 ) )