Theorem xaddnepnf 9763

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 9686	. 2 |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
2		xrnepnf 9686	. . . 4 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
3		rexadd 9757	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 7859	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2234	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renepnfd 7929	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= +oo )
7		oveq2 5833	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
8		rexr 7924	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renepnf 7926	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
10		xaddmnf1 9753	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
11	8, 9, 10	syl2anc 409	. . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
12	7, 11	sylan9eqr 2212	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
13		mnfnepnf 7934	. . . . . . 7 |- -oo =/= +oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -oo =/= +oo )
15	12, 14	eqnetrd 2351	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) =/= +oo )
16	6, 15	jaodan 787	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = -oo ) ) -> ( A +e B ) =/= +oo )
17	2, 16	sylan2b 285	. . 3 |- ( ( A e. RR /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
18		oveq1 5832	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
19		xaddmnf2 9754	. . . . 5 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
20	18, 19	sylan9eq 2210	. . . 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 2351	. . 3 |- ( ( A = -oo /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
23	17, 22	jaoian 785	. 2 |- ( ( ( A e. RR \/ A = -oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
24	1, 23	sylanb 282	1 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1335   e. wcel 2128   =/= wne 2327  (class class class)co 5825   RRcr 7732   + caddc 7736   +oocpnf 7910   -oocmnf 7911   RR*cxr 7912   +ecxad 9678

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-1re 7827  ax-addrcl 7830  ax-rnegex 7842

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-pnf 7915  df-mnf 7916  df-xr 7917  df-xadd 9681

This theorem is referenced by:  xlt2add  9785