Theorem lemulge12

Description: Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011)

	Ref	Expression
Assertion	lemulge12	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐵 · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		lemulge11	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐴 · 𝐵 ) )
2		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
4		mulcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
5	2 3 4	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
6	5	breq2d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ ( 𝐴 · 𝐵 ) ↔ 𝐴 ≤ ( 𝐵 · 𝐴 ) ) )
7	6	adantr	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → ( 𝐴 ≤ ( 𝐴 · 𝐵 ) ↔ 𝐴 ≤ ( 𝐵 · 𝐴 ) ) )
8	1 7	mpbid	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 1 ≤ 𝐵 ) ) → 𝐴 ≤ ( 𝐵 · 𝐴 ) )