Metamath Proof Explorer


Theorem lesubadd2i

Description: 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999)

Ref Expression
Hypotheses lt2.1 𝐴 ∈ ℝ
lt2.2 𝐵 ∈ ℝ
lt2.3 𝐶 ∈ ℝ
Assertion lesubadd2i ( ( 𝐴𝐵 ) ≤ 𝐶𝐴 ≤ ( 𝐵 + 𝐶 ) )

Proof

Step Hyp Ref Expression
1 lt2.1 𝐴 ∈ ℝ
2 lt2.2 𝐵 ∈ ℝ
3 lt2.3 𝐶 ∈ ℝ
4 lesubadd2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴𝐵 ) ≤ 𝐶𝐴 ≤ ( 𝐵 + 𝐶 ) ) )
5 1 2 3 4 mp3an ( ( 𝐴𝐵 ) ≤ 𝐶𝐴 ≤ ( 𝐵 + 𝐶 ) )