Theorem xnn0xr

Description: An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020)

	Ref	Expression
Assertion	xnn0xr	|- ( A e. NN0* -> A e. RR* )

Proof

Step	Hyp	Ref	Expression
1		elxnn0	|- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re	|- ( A e. NN0 -> A e. RR )
3	2	rexrd	|- ( A e. NN0 -> A e. RR* )
4		pnfxr	|- +oo e. RR*
5		eleq1	|- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
6	4 5	mpbiri	|- ( A = +oo -> A e. RR* )
7	3 6	jaoi	|- ( ( A e. NN0 \/ A = +oo ) -> A e. RR* )
8	1 7	sylbi	|- ( A e. NN0* -> A e. RR* )