Theorem ltadd1dd

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	ltadd1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltadd1dd	⊢ ( 𝜑 → ( 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		ltadd1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
5	1 2 3	ltadd1d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) < ( 𝐵 + 𝐶 ) )