Theorem ltdivgt1

Description: Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Hypotheses	ltdivgt1.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	ltdivgt1.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	ltdivgt1	⊢ ( 𝜑 → ( 1 < 𝐵 ↔ ( 𝐴 / 𝐵 ) < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltdivgt1.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		ltdivgt1.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		1rp	⊢ 1 ∈ ℝ+
4	3	a1i	⊢ ( 𝜑 → 1 ∈ ℝ+ )
5	4 2 1	ltdiv2d	⊢ ( 𝜑 → ( 1 < 𝐵 ↔ ( 𝐴 / 𝐵 ) < ( 𝐴 / 1 ) ) )
6	1	rpcnd	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
7	6	div1d	⊢ ( 𝜑 → ( 𝐴 / 1 ) = 𝐴 )
8	7	breq2d	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) < ( 𝐴 / 1 ) ↔ ( 𝐴 / 𝐵 ) < 𝐴 ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( 1 < 𝐵 ↔ ( 𝐴 / 𝐵 ) < 𝐴 ) )