Theorem lediv23d

Description: Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltdiv23d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltdiv23d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltdiv23d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	lediv23d.4	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≤ 𝐶 )
Assertion	lediv23d	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ltdiv23d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltdiv23d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltdiv23d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		lediv23d.4	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≤ 𝐶 )
5	2	rpregt0d	⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
6	3	rpregt0d	⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) )
7		lediv23	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐵 ) ≤ 𝐶 ↔ ( 𝐴 / 𝐶 ) ≤ 𝐵 ) )
8	1 5 6 7	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ≤ 𝐶 ↔ ( 𝐴 / 𝐶 ) ≤ 𝐵 ) )
9	4 8	mpbid	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) ≤ 𝐵 )