Theorem int-ineq2ndprincd

Description: SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-ineq2ndprincd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	int-ineq2ndprincd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-ineq2ndprincd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	int-ineq2ndprincd.4	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
	int-ineq2ndprincd.5	⊢ ( 𝜑 → 0 ≤ 𝐶 )
Assertion	int-ineq2ndprincd	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) ≤ ( 𝐴 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		int-ineq2ndprincd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		int-ineq2ndprincd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		int-ineq2ndprincd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		int-ineq2ndprincd.4	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
5		int-ineq2ndprincd.5	⊢ ( 𝜑 → 0 ≤ 𝐶 )
6	2 1 3 5 4	lemul1ad	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) ≤ ( 𝐴 · 𝐶 ) )