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Metamath Proof Explorer

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 ) )