Theorem xaddnepnf 9815

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 9735	. 2 |- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
2		xrnepnf 9735	. . . 4 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
3		rexadd 9809	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
4		readdcl 7900	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
5	3, 4	eqeltrd 2247	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) e. RR )
6	5	renepnfd 7970	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) =/= +oo )
7		oveq2 5861	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
8		rexr 7965	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
9		renepnf 7967	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
10		xaddmnf1 9805	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
11	8, 9, 10	syl2anc 409	. . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
12	7, 11	sylan9eqr 2225	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
13		mnfnepnf 7975	. . . . . . 7 |- -oo =/= +oo
14	13	a1i 9	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -oo =/= +oo )
15	12, 14	eqnetrd 2364	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) =/= +oo )
16	6, 15	jaodan 792	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = -oo ) ) -> ( A +e B ) =/= +oo )
17	2, 16	sylan2b 285	. . 3 |- ( ( A e. RR /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
18		oveq1 5860	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
19		xaddmnf2 9806	. . . . 5 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
20	18, 19	sylan9eq 2223	. . . 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 2364	. . 3 |- ( ( A = -oo /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
23	17, 22	jaoian 790	. 2 |- ( ( ( A e. RR \/ A = -oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
24	1, 23	sylanb 282	1 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   =/= wne 2340  (class class class)co 5853   RRcr 7773   + caddc 7777   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   +ecxad 9727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-rnegex 7883

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-xadd 9730

This theorem is referenced by:  xlt2add  9837