Theorem sgnp

Description: The signum of a positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015)

	Ref	Expression
Assertion	sgnp	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \mathrm{sgn}\left(A\right)=1$$

Proof

Step	Hyp	Ref	Expression
1		sgnval	$$⊢A\in {ℝ}^{*}\to \mathrm{sgn}\left(A\right)=if\left(A=0,0,if\left(A<0,-1,1\right)\right)$$
2	1	adantr	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \mathrm{sgn}\left(A\right)=if\left(A=0,0,if\left(A<0,-1,1\right)\right)$$
3		0xr	$$⊢0\in {ℝ}^{*}$$
4		xrltne	$$⊢\left(0\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\wedge 0<A\right)\to A\ne 0$$
5	3 4	mp3an1	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to A\ne 0$$
6	5	neneqd	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \neg A=0$$
7	6	iffalsed	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to if\left(A=0,0,if\left(A<0,-1,1\right)\right)=if\left(A<0,-1,1\right)$$
8		xrltnsym	$$⊢\left(0\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\right)\to \left(0<A\to \neg A<0\right)$$
9	3 8	mpan	$$⊢A\in {ℝ}^{*}\to \left(0<A\to \neg A<0\right)$$
10	9	imp	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \neg A<0$$
11	10	iffalsed	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to if\left(A<0,-1,1\right)=1$$
12	2 7 11	3eqtrd	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \mathrm{sgn}\left(A\right)=1$$