Theorem ltmuldiv2

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004)

	Ref	Expression
Assertion	ltmuldiv2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐶 · 𝐴 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		recn	⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
3		mulcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
5	4	adantrr	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
6	5	3adant2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
7	6	breq1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ ( 𝐶 · 𝐴 ) < 𝐵 ) )
8		ltmuldiv	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 · 𝐶 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )
9	7 8	bitr3d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐶 · 𝐴 ) < 𝐵 ↔ 𝐴 < ( 𝐵 / 𝐶 ) ) )