Theorem ltmul12ad

Description: Comparison of product of two positive 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 )
	ltmul12ad.4	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	ltmul12ad.5	⊢ ( 𝜑 → 𝐴 < 𝐵 )
	ltmul12ad.6	⊢ ( 𝜑 → 0 ≤ 𝐶 )
	ltmul12ad.7	⊢ ( 𝜑 → 𝐶 < 𝐷 )
Assertion	ltmul12ad	⊢ ( 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		ltmul12ad.4	⊢ ( 𝜑 → 0 ≤ 𝐴 )
6		ltmul12ad.5	⊢ ( 𝜑 → 𝐴 < 𝐵 )
7		ltmul12ad.6	⊢ ( 𝜑 → 0 ≤ 𝐶 )
8		ltmul12ad.7	⊢ ( 𝜑 → 𝐶 < 𝐷 )
9	1 2	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
10	5 6	jca	⊢ ( 𝜑 → ( 0 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) )
11	3 4	jca	⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) )
12	7 8	jca	⊢ ( 𝜑 → ( 0 ≤ 𝐶 ∧ 𝐶 < 𝐷 ) )
13		ltmul12a	⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) ) ∧ ( ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ∧ ( 0 ≤ 𝐶 ∧ 𝐶 < 𝐷 ) ) ) → ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐷 ) )
14	9 10 11 12 13	syl22anc	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) < ( 𝐵 · 𝐷 ) )