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Metamath Proof Explorer

Theorem lemulge11d

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

Ref Expression
Hypotheses ltp1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
divgt0d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
lemulge11d.3 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
lemulge11d.4 ⊢ ( 𝜑 → 1 ≤ 𝐵 )
Assertion lemulge11d ( 𝜑 → 𝐴 ≤ ( 𝐴 · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ltp1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 divgt0d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 lemulge11d.3 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
4 lemulge11d.4 ⊢ ( 𝜑 → 1 ≤ 𝐵 )
5 lemulge11 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐴 · 𝐵 ) )
6 1 2 3 4 5 syl22anc ⊢ ( 𝜑 → 𝐴 ≤ ( 𝐴 · 𝐵 ) )