Theorem xrge0plusg

Description: The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017)

	Ref	Expression
Assertion	xrge0plusg	$$⊢{+}_{𝑒}={+}_{\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)}$$

Proof

Step	Hyp	Ref	Expression
1		ovex	$$⊢\left[0,\mathrm{+\infty }\right]\in \mathrm{V}$$
2		eqid	$$⊢{ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]={ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]$$
3		xrsadd	$$⊢{+}_{𝑒}={+}_{{ℝ}_{𝑠}^{*}}$$
4	2 3	ressplusg	$$⊢\left[0,\mathrm{+\infty }\right]\in \mathrm{V}\to {+}_{𝑒}={+}_{\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)}$$
5	1 4	ax-mp	$$⊢{+}_{𝑒}={+}_{\left({ℝ}_{𝑠}^{*}{\upharpoonright }_{𝑠}\left[0,\mathrm{+\infty }\right]\right)}$$