This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem le2addi

Description: Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999)

Ref Expression
Hypotheses lt2.1 ⊢ 𝐴 ∈ ℝ
lt2.2 ⊢ 𝐵 ∈ ℝ
lt2.3 ⊢ 𝐶 ∈ ℝ
lt.4 ⊢ 𝐷 ∈ ℝ
Assertion le2addi ( ( 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 le2add ⊢ ( ( ( 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 ) )