Theorem xadd0ge2

Description: A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xadd0ge2.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xadd0ge2.b	⊢ ( 𝜑 → 𝐵 ∈ ( 0 [,] +∞ ) )
Assertion	xadd0ge2	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐵 +𝑒 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xadd0ge2.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xadd0ge2.b	⊢ ( 𝜑 → 𝐵 ∈ ( 0 [,] +∞ ) )
3	1 2	xadd0ge	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐴 +𝑒 𝐵 ) )
4		iccssxr	⊢ ( 0 [,] +∞ ) ⊆ ℝ*
5	4 2	sseldi	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
6	1 5	xaddcomd	⊢ ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) = ( 𝐵 +𝑒 𝐴 ) )
7	3 6	breqtrd	⊢ ( 𝜑 → 𝐴 ≤ ( 𝐵 +𝑒 𝐴 ) )