Theorem xaddnepnf 9877

Description: Closure of extended real addition in the subset RR* / { +oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddnepnf	|- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )

Proof of Theorem xaddnepnf

Step	Hyp	Ref	Expression
1		xrnepnf 9797	. 2 |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
2		xrnepnf 9797	. . . 4 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
3		rexadd 9871	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 7956	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2266	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renepnfd 8027	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= +oo )
7		oveq2 5899	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
8		rexr 8022	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renepnf 8024	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
10		xaddmnf1 9867	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
11	8, 9, 10	syl2anc 411	. . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
12	7, 11	sylan9eqr 2244	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
13		mnfnepnf 8032	. . . . . . 7 |- -oo =/= +oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -oo =/= +oo )
15	12, 14	eqnetrd 2384	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) =/= +oo )
16	6, 15	jaodan 798	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = -oo ) ) -> ( A +e B ) =/= +oo )
17	2, 16	sylan2b 287	. . 3 |- ( ( A e. RR /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
18		oveq1 5898	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
19		xaddmnf2 9868	. . . . 5 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
20	18, 19	sylan9eq 2242	. . . 4 |- ( ( A = -oo /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) = -oo )
21	13	a1i 9	. . . 4 |- ( ( A = -oo /\ ( B e. RR* /\ B =/= +oo ) ) -> -oo =/= +oo )
22	20, 21	eqnetrd 2384	. . 3 |- ( ( A = -oo /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
23	17, 22	jaoian 796	. 2 |- ( ( ( A e. RR \/ A = -oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
24	1, 23	sylanb 284	1 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2160   =/= wne 2360  (class class class)co 5891   RRcr 7829   + caddc 7833   +oocpnf 8008   -oocmnf 8009   RR*cxr 8010   +ecxad 9789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927  ax-rnegex 7939

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-xadd 9792

This theorem is referenced by:  xlt2add  9899