Theorem lemul12bd

Description: Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	divgt0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	lemul1ad.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	ltmul12ad.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	lemul12bd.4	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	lemul12bd.5	⊢ ( 𝜑 → 0 ≤ 𝐷 )
	lemul12bd.6	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	lemul12bd.7	⊢ ( 𝜑 → 𝐶 ≤ 𝐷 )
Assertion	lemul12bd	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		divgt0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lemul1ad.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		ltmul12ad.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		lemul12bd.4	⊢ ( 𝜑 → 0 ≤ 𝐴 )
6		lemul12bd.5	⊢ ( 𝜑 → 0 ≤ 𝐷 )
7		lemul12bd.6	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
8		lemul12bd.7	⊢ ( 𝜑 → 𝐶 ≤ 𝐷 )
9	1 5	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
10	4 6	jca	⊢ ( 𝜑 → ( 𝐷 ∈ ℝ ∧ 0 ≤ 𝐷 ) )
11		lemul12b	⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ ( 𝐷 ∈ ℝ ∧ 0 ≤ 𝐷 ) ) ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷 ) → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐷 ) ) )
12	9 2 3 10 11	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷 ) → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐷 ) ) )
13	7 8 12	mp2and	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐷 ) )