Theorem leadd2i

Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
	lt2.3	⊢ 𝐶 ∈ ℝ
Assertion	leadd2i	⊢ ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		lt2.3	⊢ 𝐶 ∈ ℝ
4		leadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) ) )
5	1 2 3 4	mp3an	⊢ ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 + 𝐴 ) ≤ ( 𝐶 + 𝐵 ) )