Theorem xaddnepnf 10092

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 10012	. 2 |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
2		xrnepnf 10012	. . . 4 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
3		rexadd 10086	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 8157	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2308	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renepnfd 8229	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= +oo )
7		oveq2 6025	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
8		rexr 8224	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renepnf 8226	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
10		xaddmnf1 10082	. . . . . . . 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 2286	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
13		mnfnepnf 8234	. . . . . . 7 |- -oo =/= +oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -oo =/= +oo )
15	12, 14	eqnetrd 2426	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) =/= +oo )
16	6, 15	jaodan 804	. . . 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 6024	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
19		xaddmnf2 10083	. . . . 5 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
20	18, 19	sylan9eq 2284	. . . 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 2426	. . 3 |- ( ( A = -oo /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
23	17, 22	jaoian 802	. 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 715   = wceq 1397   e. wcel 2202   =/= wne 2402  (class class class)co 6017   RRcr 8030   + caddc 8034   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   +ecxad 10004

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-rnegex 8140

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-xadd 10007

This theorem is referenced by:  xlt2add  10114