Theorem xaddnemnf 12067

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 11951	. 2 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
2		xrnemnf 11951	. . . 4 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
3		rexadd 12063	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 10019	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2701	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renemnfd 10091	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= -oo )
7		oveq2 6658	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
8		rexr 10085	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renemnf 10088	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
10		xaddpnf1 12057	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
11	8, 9, 10	syl2anc 693	. . . . . . 7 |- ( A e. RR -> ( A +e +oo ) = +oo )
12	7, 11	sylan9eqr 2678	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
13		pnfnemnf 10094	. . . . . . 7 |- +oo =/= -oo
14	13	a1i 11	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> +oo =/= -oo )
15	12, 14	eqnetrd 2861	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) =/= -oo )
16	6, 15	jaodan 826	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo ) ) -> ( A +e B ) =/= -oo )
17	2, 16	sylan2b 492	. . 3 |- ( ( A e. RR /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
18		oveq1 6657	. . . . 5 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
19		xaddpnf2 12058	. . . . 5 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
20	18, 19	sylan9eq 2676	. . . 4 |- ( ( A = +oo /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) = +oo )
21	13	a1i 11	. . . 4 |- ( ( A = +oo /\ ( B e. RR* /\ B =/= -oo ) ) -> +oo =/= -oo )
22	20, 21	eqnetrd 2861	. . 3 |- ( ( A = +oo /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
23	17, 22	jaoian 824	. 2 |- ( ( ( A e. RR \/ A = +oo ) /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
24	1, 23	sylanb 489	1 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   RRcr 9935   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   +ecxad 11944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-xadd 11947

This theorem is referenced by:  xaddass  12079  xlt2add  12090  xadd4d  12133  xrs1mnd  19784