Theorem xle0neg1

Description: Extended real version of le0neg1 . (Contributed by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	xle0neg1	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ≤ 0 ↔ 0 ≤ -𝑒 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		xleneg	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐴 ≤ 0 ↔ -𝑒 0 ≤ -𝑒 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ≤ 0 ↔ -𝑒 0 ≤ -𝑒 𝐴 ) )
4		xneg0	⊢ -𝑒 0 = 0
5	4	breq1i	⊢ ( -𝑒 0 ≤ -𝑒 𝐴 ↔ 0 ≤ -𝑒 𝐴 )
6	3 5	bitrdi	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ≤ 0 ↔ 0 ≤ -𝑒 𝐴 ) )