Theorem xle0neg2

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

	Ref	Expression
Assertion	xle0neg2	$$⊢A\in {ℝ}^{*}\to \left(0\le A\leftrightarrow -A\le 0\right)$$

Proof

Step	Hyp	Ref	Expression
1		0xr	$$⊢0\in {ℝ}^{*}$$
2		xleneg	$$⊢\left(0\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\right)\to \left(0\le A\leftrightarrow -A\le -0\right)$$
3	1 2	mpan	$$⊢A\in {ℝ}^{*}\to \left(0\le A\leftrightarrow -A\le -0\right)$$
4		xneg0	$$⊢-0=0$$
5	4	breq2i	$$⊢-A\le -0\leftrightarrow -A\le 0$$
6	3 5	syl6bb	$$⊢A\in {ℝ}^{*}\to \left(0\le A\leftrightarrow -A\le 0\right)$$