Theorem ltadd1i

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by NM, 21-Jan-1997)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
	lt2.3	⊢ 𝐶 ∈ ℝ
Assertion	ltadd1i	⊢ ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		lt2.3	⊢ 𝐶 ∈ ℝ
4		ltadd1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) )