Theorem xle0neg2

Description: Extended real version of le0neg2 . (Contributed by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	xle0neg2	⊢ ( 𝐴 ∈ ℝ* → ( 0 ≤ 𝐴 ↔ -𝑒 𝐴 ≤ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		xleneg	⊢ ( ( 0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 0 ≤ 𝐴 ↔ -𝑒 𝐴 ≤ -𝑒 0 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ* → ( 0 ≤ 𝐴 ↔ -𝑒 𝐴 ≤ -𝑒 0 ) )
4		xneg0	⊢ -𝑒 0 = 0
5	4	breq2i	⊢ ( -𝑒 𝐴 ≤ -𝑒 0 ↔ -𝑒 𝐴 ≤ 0 )
6	3 5	bitrdi	⊢ ( 𝐴 ∈ ℝ* → ( 0 ≤ 𝐴 ↔ -𝑒 𝐴 ≤ 0 ) )