Theorem lemulge12d

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	lemulge12d	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐵 · 𝐴 ) )

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		lemulge12	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐵 · 𝐴 ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐵 · 𝐴 ) )