Theorem xaddnepnf 9933

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 9853	. 2 |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
2		xrnepnf 9853	. . . 4 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
3		rexadd 9927	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 8005	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2273	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renepnfd 8077	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= +oo )
7		oveq2 5930	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
8		rexr 8072	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renepnf 8074	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
10		xaddmnf1 9923	. . . . . . . 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 2251	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
13		mnfnepnf 8082	. . . . . . 7 |- -oo =/= +oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -oo =/= +oo )
15	12, 14	eqnetrd 2391	. . . . 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 5929	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
19		xaddmnf2 9924	. . . . 5 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
20	18, 19	sylan9eq 2249	. . . 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 2391	. . 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 2167   =/= wne 2367  (class class class)co 5922   RRcr 7878   + caddc 7882   +oocpnf 8058   -oocmnf 8059   RR*cxr 8060   +ecxad 9845

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976  ax-rnegex 7988

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-xadd 9848

This theorem is referenced by:  xlt2add  9955