Theorem ltadd1d

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	ltadd1d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		ltadd1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) ) )