Theorem ltmul1i

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (Contributed by NM, 16-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
	ltmul1.3	⊢ 𝐶 ∈ ℝ
Assertion	ltmul1i	⊢ ( 0 < 𝐶 → ( 𝐴 < 𝐵 ↔ ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐶 ) ) )

Proof

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