Theorem xaddnemnf 10053

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

Assertion

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

Proof of Theorem xaddnemnf

Step	Hyp	Ref	Expression
1		xrnemnf 9973	. 2 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
2		xrnemnf 9973	. . . 4 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
3		rexadd 10048	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 8125	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2306	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renemnfd 8198	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= -oo )
7		oveq2 6009	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
8		rexr 8192	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renemnf 8195	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
10		xaddpnf1 10042	. . . . . . . 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 2284	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
13		pnfnemnf 8201	. . . . . . 7 |- +oo =/= -oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> +oo =/= -oo )
15	12, 14	eqnetrd 2424	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) =/= -oo )
16	6, 15	jaodan 802	. . . 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 6008	. . . . 5 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
19		xaddpnf2 10043	. . . . 5 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
20	18, 19	sylan9eq 2282	. . . 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 2424	. . 3 |- ( ( A = +oo /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
23	17, 22	jaoian 800	. 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 713   = wceq 1395   e. wcel 2200   =/= wne 2400  (class class class)co 6001   RRcr 7998   + caddc 8002   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180   +ecxad 9966

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-rnegex 8108

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-xadd 9969

This theorem is referenced by:  xaddass  10065  xlt2add  10076  xadd4d  10081  xleaddadd  10083