Theorem ledivmul2

Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005)

	Ref	Expression
Assertion	ledivmul2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 · 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ledivmul	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐶 · 𝐵 ) ) )
2		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
3		recn	⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℂ )
4		mulcom	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
5	2 3 4	syl2an	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
6	5	adantrr	⊢ ( ( 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
7	6	3adant1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
8	7	breq2d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ ( 𝐵 · 𝐶 ) ↔ 𝐴 ≤ ( 𝐶 · 𝐵 ) ) )
9	1 8	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( ( 𝐴 / 𝐶 ) ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 · 𝐶 ) ) )