Theorem xnn0xr 9203

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

Assertion

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

Proof of Theorem xnn0xr

Step	Hyp	Ref	Expression
1		elxnn0 9200	. 2 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2		nn0re 9144	. . . 4 |- ( A e. NN0 -> A e. RR )
3	2	rexrd 7969	. . 3 |- ( A e. NN0 -> A e. RR* )
4		pnfxr 7972	. . . 4 |- +oo e. RR*
5		eleq1 2233	. . . 4 |- ( A = +oo -> ( A e. RR* <-> +oo e. RR* ) )
6	4, 5	mpbiri 167	. . 3 |- ( A = +oo -> A e. RR* )
7	3, 6	jaoi 711	. 2 |- ( ( A e. NN0 \/ A = +oo ) -> A e. RR* )
8	1, 7	sylbi 120	1 |- ( A e. NN0* -> A e. RR* )

Syntax hints:   -> wi 4   \/ wo 703   = wceq 1348   e. wcel 2141   +oocpnf 7951   RR*cxr 7953   NN0cn0 9135  NN0*cxnn0 9198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-rnegex 7883

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-pnf 7956  df-xr 7958  df-inn 8879  df-n0 9136  df-xnn0 9199

This theorem is referenced by:  xnn0xrnemnf  9210  xnn0dcle  9759  xnn0letri  9760