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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