Theorem addge02

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005)

	Ref	Expression
Assertion	addge02	⊢ ( ( 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		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
4		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
5	2 3 4	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6	5	breq2d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ ( 𝐴 + 𝐵 ) ↔ 𝐴 ≤ ( 𝐵 + 𝐴 ) ) )
7	1 6	bitrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 + 𝐴 ) ) )