Theorem lt2addi

Description: Adding both side of two inequalities. Theorem I.25 of Apostol p. 20. (Contributed by NM, 14-May-1999)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
	lt2.3	⊢ 𝐶 ∈ ℝ
	lt.4	⊢ 𝐷 ∈ ℝ
Assertion	lt2addi	⊢ ( ( 𝐴 < 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		lt2.3	⊢ 𝐶 ∈ ℝ
4		lt.4	⊢ 𝐷 ∈ ℝ
5		lt2add	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ) → ( ( 𝐴 < 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) ) )
6	1 2 3 4 5	mp4an	⊢ ( ( 𝐴 < 𝐶 ∧ 𝐵 < 𝐷 ) → ( 𝐴 + 𝐵 ) < ( 𝐶 + 𝐷 ) )