Theorem subge02

Description: Nonnegative subtraction. (Contributed by NM, 27-Jul-2005)

	Ref	Expression
Assertion	subge02	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 − 𝐵 ) ≤ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		addge01	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) )
2		lesubadd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐴 ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) )
3	2	3anidm13	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐴 ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) )
4	1 3	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 − 𝐵 ) ≤ 𝐴 ) )