Theorem ltmulgt12

Description: Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005)

	Ref	Expression
Assertion	ltmulgt12	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐵 ↔ 𝐴 < ( 𝐵 · 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltmulgt11	⊢ ( ( 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	3adant3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
7	6	breq2d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 < ( 𝐴 · 𝐵 ) ↔ 𝐴 < ( 𝐵 · 𝐴 ) ) )
8	1 7	bitrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < 𝐵 ↔ 𝐴 < ( 𝐵 · 𝐴 ) ) )