Theorem ltadd2i

Description: Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	lt.2	⊢ 𝐵 ∈ ℝ
	lt.3	⊢ 𝐶 ∈ ℝ
Assertion	ltadd2i	⊢ ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		lt.2	⊢ 𝐵 ∈ ℝ
3		lt.3	⊢ 𝐶 ∈ ℝ
4		ltadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )
5	1 2 3 4	mp3an	⊢ ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) )