Theorem leaddsub2d

Description: 'Less than or equal to' relationship between and addition and subtraction. (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	leaddsub2d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltadd1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		leaddsub2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 + 𝐵 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 − 𝐴 ) ) )