Theorem lesubadd2d

Description: 'Less than or equal to' relationship between subtraction and addition. (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	lesubadd2d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 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		lesubadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 𝐵 + 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 𝐵 + 𝐶 ) ) )