Metamath Proof Explorer


Theorem lemuldiv4d

Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020)

Ref Expression
Hypotheses lemuldiv4d.1 ( 𝜑𝐵 ≤ ( 𝐶 / 𝐴 ) )
lemuldiv4d.2 ( 𝜑 → 0 < 𝐴 )
lemuldiv4d.3 ( 𝜑𝐴 ∈ ℝ )
lemuldiv4d.4 ( 𝜑𝐵 ∈ ℝ )
lemuldiv4d.5 ( 𝜑𝐶 ∈ ℝ )
Assertion lemuldiv4d ( 𝜑 → ( 𝐵 · 𝐴 ) ≤ 𝐶 )

Proof

Step Hyp Ref Expression
1 lemuldiv4d.1 ( 𝜑𝐵 ≤ ( 𝐶 / 𝐴 ) )
2 lemuldiv4d.2 ( 𝜑 → 0 < 𝐴 )
3 lemuldiv4d.3 ( 𝜑𝐴 ∈ ℝ )
4 lemuldiv4d.4 ( 𝜑𝐵 ∈ ℝ )
5 lemuldiv4d.5 ( 𝜑𝐶 ∈ ℝ )
6 lemuldiv ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( ( 𝐵 · 𝐴 ) ≤ 𝐶𝐵 ≤ ( 𝐶 / 𝐴 ) ) )
7 4 5 3 2 6 syl112anc ( 𝜑 → ( ( 𝐵 · 𝐴 ) ≤ 𝐶𝐵 ≤ ( 𝐶 / 𝐴 ) ) )
8 7 bicomd ( 𝜑 → ( 𝐵 ≤ ( 𝐶 / 𝐴 ) ↔ ( 𝐵 · 𝐴 ) ≤ 𝐶 ) )
9 1 8 mpbid ( 𝜑 → ( 𝐵 · 𝐴 ) ≤ 𝐶 )