Theorem leadd1dd

Description: Addition to both sides of 'less than or equal to'. (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 )
	leadd1dd.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion	leadd1dd	⊢ ( 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		leadd1dd.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5	1 2 3	leadd1d	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 + 𝐶 ) ≤ ( 𝐵 + 𝐶 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) ≤ ( 𝐵 + 𝐶 ) )