Theorem xaddnemnf 9923

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 9843	. 2 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
2		xrnemnf 9843	. . . 4 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
3		rexadd 9918	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 7998	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2270	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renemnfd 8071	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= -oo )
7		oveq2 5926	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
8		rexr 8065	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renemnf 8068	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
10		xaddpnf1 9912	. . . . . . . 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 2248	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
13		pnfnemnf 8074	. . . . . . 7 |- +oo =/= -oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> +oo =/= -oo )
15	12, 14	eqnetrd 2388	. . . . 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 5925	. . . . 5 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
19		xaddpnf2 9913	. . . . 5 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
20	18, 19	sylan9eq 2246	. . . 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 2388	. . 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 2164   =/= wne 2364  (class class class)co 5918   RRcr 7871   + caddc 7875   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   +ecxad 9836

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-rnegex 7981

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-xadd 9839

This theorem is referenced by:  xaddass  9935  xlt2add  9946  xadd4d  9951  xleaddadd  9953