Theorem sgndm

Description: The domain of the signum function. (Contributed by AV, 16-Jun-2026)

	Ref	Expression
Assertion	sgndm	$$⊢\mathrm{dom}\mathrm{sgn}={ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		c0ex	$$⊢0\in \mathrm{V}$$
2		negex	$$⊢-1\in \mathrm{V}$$
3		1ex	$$⊢1\in \mathrm{V}$$
4	2 3	ifex	$$⊢if\left(x<0,-1,1\right)\in \mathrm{V}$$
5	1 4	ifex	$$⊢if\left(x=0,0,if\left(x<0,-1,1\right)\right)\in \mathrm{V}$$
6		df-sgn	$$⊢\mathrm{sgn}=\left(x\in {ℝ}^{*}⟼if\left(x=0,0,if\left(x<0,-1,1\right)\right)\right)$$
7	5 6	dmmpti	$$⊢\mathrm{dom}\mathrm{sgn}={ℝ}^{*}$$