Theorem ltmuldivi

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
	ltmul1.3	⊢ 𝐶 ∈ ℝ
Assertion	ltmuldivi	⊢ ( 0 < 𝐶 → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		ltmul1.3	⊢ 𝐶 ∈ ℝ
4		ltmuldiv	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )
5	1 2 4	mp3an12	⊢ ( ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )
6	3 5	mpan	⊢ ( 0 < 𝐶 → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )