Theorem lt2addd

Description: Adding both side of two inequalities. Theorem I.25 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	lt2addd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	lt2addd.5	⊢ ( 𝜑 → 𝐴 < 𝐶 )
	lt2addd.6	⊢ ( 𝜑 → 𝐵 < 𝐷 )
Assertion	lt2addd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		lt2addd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		lt2addd.5	⊢ ( 𝜑 → 𝐴 < 𝐶 )
6		lt2addd.6	⊢ ( 𝜑 → 𝐵 < 𝐷 )
7	2 4 6	ltled	⊢ ( 𝜑 → 𝐵 ≤ 𝐷 )
8	1 2 3 4 5 7	ltleaddd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) )