Theorem xaddeq0 29518

Description: Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)

Assertion

Ref	Expression
xaddeq0	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A +e B ) = 0 <-> A = -e B ) )

Proof of Theorem xaddeq0

Step	Hyp	Ref	Expression
1		elxr 11950	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		simpll 790	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A e. RR )
3	2	rexrd 10089	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A e. RR* )
4		xnegneg 12045	. . . . . . 7 |- ( A e. RR* -> -e -e A = A )
5	3, 4	syl 17	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -e -e A = A )
6	3	xnegcld 12130	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -e A e. RR* )
7		xaddid2 12073	. . . . . . . . 9 |- ( -e A e. RR* -> ( 0 +e -e A ) = -e A )
8	6, 7	syl 17	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( 0 +e -e A ) = -e A )
9		simplr 792	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> B e. RR* )
10		xaddcom 12071	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
11	3, 9, 10	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( A +e B ) = ( B +e A ) )
12	11	oveq1d 6665	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( ( A +e B ) +e -e A ) = ( ( B +e A ) +e -e A ) )
13		simpr 477	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( A +e B ) = 0 )
14	13	oveq1d 6665	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( ( A +e B ) +e -e A ) = ( 0 +e -e A ) )
15		xpncan 12081	. . . . . . . . . . 11 |- ( ( B e. RR* /\ A e. RR ) -> ( ( B +e A ) +e -e A ) = B )
16	15	ancoms 469	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR* ) -> ( ( B +e A ) +e -e A ) = B )
17	16	adantr 481	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( ( B +e A ) +e -e A ) = B )
18	12, 14, 17	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( 0 +e -e A ) = B )
19	8, 18	eqtr3d 2658	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -e A = B )
20		xnegeq 12038	. . . . . . 7 |- ( -e A = B -> -e -e A = -e B )
21	19, 20	syl 17	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -e -e A = -e B )
22	5, 21	eqtr3d 2658	. . . . 5 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A = -e B )
23	22	ex 450	. . . 4 |- ( ( A e. RR /\ B e. RR* ) -> ( ( A +e B ) = 0 -> A = -e B ) )
24		simpll 790	. . . . . 6 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A = +oo )
25		simplr 792	. . . . . . . . . 10 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> B e. RR* )
26	24	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( A +e B ) = ( +oo +e B ) )
27		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( A +e B ) = 0 )
28	26, 27	eqtr3d 2658	. . . . . . . . . . . 12 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( +oo +e B ) = 0 )
29		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
30		renepnf 10087	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= +oo )
31	29, 30	mp1i 13	. . . . . . . . . . . 12 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> 0 =/= +oo )
32	28, 31	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( +oo +e B ) =/= +oo )
33	32	neneqd 2799	. . . . . . . . . 10 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -. ( +oo +e B ) = +oo )
34		xaddpnf2 12058	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
35	34	stoic1a 1697	. . . . . . . . . 10 |- ( ( B e. RR* /\ -. ( +oo +e B ) = +oo ) -> -. B =/= -oo )
36	25, 33, 35	syl2anc 693	. . . . . . . . 9 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -. B =/= -oo )
37		nne 2798	. . . . . . . . 9 |- ( -. B =/= -oo <-> B = -oo )
38	36, 37	sylib 208	. . . . . . . 8 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> B = -oo )
39		xnegeq 12038	. . . . . . . 8 |- ( B = -oo -> -e B = -e -oo )
40	38, 39	syl 17	. . . . . . 7 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -e B = -e -oo )
41		xnegmnf 12041	. . . . . . 7 |- -e -oo = +oo
42	40, 41	syl6req 2673	. . . . . 6 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> +oo = -e B )
43	24, 42	eqtrd 2656	. . . . 5 |- ( ( ( A = +oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A = -e B )
44	43	ex 450	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( ( A +e B ) = 0 -> A = -e B ) )
45		simpll 790	. . . . . 6 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A = -oo )
46		simplr 792	. . . . . . . . . 10 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> B e. RR* )
47	45	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( A +e B ) = ( -oo +e B ) )
48		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( A +e B ) = 0 )
49	47, 48	eqtr3d 2658	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( -oo +e B ) = 0 )
50		renemnf 10088	. . . . . . . . . . . . 13 |- ( 0 e. RR -> 0 =/= -oo )
51	29, 50	mp1i 13	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> 0 =/= -oo )
52	49, 51	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> ( -oo +e B ) =/= -oo )
53	52	neneqd 2799	. . . . . . . . . 10 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -. ( -oo +e B ) = -oo )
54		xaddmnf2 12060	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
55	54	stoic1a 1697	. . . . . . . . . 10 |- ( ( B e. RR* /\ -. ( -oo +e B ) = -oo ) -> -. B =/= +oo )
56	46, 53, 55	syl2anc 693	. . . . . . . . 9 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -. B =/= +oo )
57		nne 2798	. . . . . . . . 9 |- ( -. B =/= +oo <-> B = +oo )
58	56, 57	sylib 208	. . . . . . . 8 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> B = +oo )
59		xnegeq 12038	. . . . . . . 8 |- ( B = +oo -> -e B = -e +oo )
60	58, 59	syl 17	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -e B = -e +oo )
61		xnegpnf 12040	. . . . . . 7 |- -e +oo = -oo
62	60, 61	syl6req 2673	. . . . . 6 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> -oo = -e B )
63	45, 62	eqtrd 2656	. . . . 5 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( A +e B ) = 0 ) -> A = -e B )
64	63	ex 450	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( ( A +e B ) = 0 -> A = -e B ) )
65	23, 44, 64	3jaoian 1393	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( ( A +e B ) = 0 -> A = -e B ) )
66	1, 65	sylanb 489	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A +e B ) = 0 -> A = -e B ) )
67		simpr 477	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> A = -e B )
68	67	oveq1d 6665	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> ( A +e B ) = ( -e B +e B ) )
69		xnegcl 12044	. . . . . 6 |- ( B e. RR* -> -e B e. RR* )
70	69	ad2antlr 763	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> -e B e. RR* )
71		simplr 792	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> B e. RR* )
72		xaddcom 12071	. . . . 5 |- ( ( -e B e. RR* /\ B e. RR* ) -> ( -e B +e B ) = ( B +e -e B ) )
73	70, 71, 72	syl2anc 693	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> ( -e B +e B ) = ( B +e -e B ) )
74		xnegid 12069	. . . . 5 |- ( B e. RR* -> ( B +e -e B ) = 0 )
75	74	ad2antlr 763	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> ( B +e -e B ) = 0 )
76	68, 73, 75	3eqtrd 2660	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A = -e B ) -> ( A +e B ) = 0 )
77	76	ex 450	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A = -e B -> ( A +e B ) = 0 ) )
78	66, 77	impbid 202	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A +e B ) = 0 <-> A = -e B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   RRcr 9935   0cc0 9936   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   -ecxne 11943   +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-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

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-reu 2919  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-sub 10268  df-neg 10269  df-xneg 11946  df-xadd 11947

This theorem is referenced by:  xrsinvgval  29677