Theorem xnegdi 12078

Description: Extended real version of xnegdi 12078. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

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

Proof of Theorem xnegdi

Step	Hyp	Ref	Expression
1		elxr 11950	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 11950	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 10026	. . . . . . . 8 |- ( A e. RR -> A e. CC )
4		recn 10026	. . . . . . . 8 |- ( B e. RR -> B e. CC )
5		negdi 10338	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
6	3, 4, 5	syl2an 494	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u ( A + B ) = ( -u A + -u B ) )
7		readdcl 10019	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexneg 12042	. . . . . . . 8 |- ( ( A + B ) e. RR -> -e ( A + B ) = -u ( A + B ) )
9	7, 8	syl 17	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = -u ( A + B ) )
10		renegcl 10344	. . . . . . . 8 |- ( A e. RR -> -u A e. RR )
11		renegcl 10344	. . . . . . . 8 |- ( B e. RR -> -u B e. RR )
12		rexadd 12063	. . . . . . . 8 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
13	10, 11, 12	syl2an 494	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
14	6, 9, 13	3eqtr4d 2666	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = ( -u A +e -u B ) )
15		rexadd 12063	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16		xnegeq 12038	. . . . . . 7 |- ( ( A +e B ) = ( A + B ) -> -e ( A +e B ) = -e ( A + B ) )
17	15, 16	syl 17	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = -e ( A + B ) )
18		rexneg 12042	. . . . . . 7 |- ( A e. RR -> -e A = -u A )
19		rexneg 12042	. . . . . . 7 |- ( B e. RR -> -e B = -u B )
20	18, 19	oveqan12d 6669	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -e A +e -e B ) = ( -u A +e -u B ) )
21	14, 17, 20	3eqtr4d 2666	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = ( -e A +e -e B ) )
22		xnegpnf 12040	. . . . . 6 |- -e +oo = -oo
23		oveq2 6658	. . . . . . . 8 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
24		rexr 10085	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
25		renemnf 10088	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
26		xaddpnf1 12057	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
27	24, 25, 26	syl2anc 693	. . . . . . . 8 |- ( A e. RR -> ( A +e +oo ) = +oo )
28	23, 27	sylan9eqr 2678	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
29		xnegeq 12038	. . . . . . 7 |- ( ( A +e B ) = +oo -> -e ( A +e B ) = -e +oo )
30	28, 29	syl 17	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = -e +oo )
31		xnegeq 12038	. . . . . . . . 9 |- ( B = +oo -> -e B = -e +oo )
32	31, 22	syl6eq 2672	. . . . . . . 8 |- ( B = +oo -> -e B = -oo )
33	32	oveq2d 6666	. . . . . . 7 |- ( B = +oo -> ( -e A +e -e B ) = ( -e A +e -oo ) )
34	18, 10	eqeltrd 2701	. . . . . . . 8 |- ( A e. RR -> -e A e. RR )
35		rexr 10085	. . . . . . . . 9 |- ( -e A e. RR -> -e A e. RR* )
36		renepnf 10087	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= +oo )
37		xaddmnf1 12059	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -e A +e -oo ) = -oo )
38	35, 36, 37	syl2anc 693	. . . . . . . 8 |- ( -e A e. RR -> ( -e A +e -oo ) = -oo )
39	34, 38	syl 17	. . . . . . 7 |- ( A e. RR -> ( -e A +e -oo ) = -oo )
40	33, 39	sylan9eqr 2678	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( -e A +e -e B ) = -oo )
41	22, 30, 40	3eqtr4a 2682	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
42		xnegmnf 12041	. . . . . 6 |- -e -oo = +oo
43		oveq2 6658	. . . . . . . 8 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
44		renepnf 10087	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
45		xaddmnf1 12059	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
46	24, 44, 45	syl2anc 693	. . . . . . . 8 |- ( A e. RR -> ( A +e -oo ) = -oo )
47	43, 46	sylan9eqr 2678	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
48		xnegeq 12038	. . . . . . 7 |- ( ( A +e B ) = -oo -> -e ( A +e B ) = -e -oo )
49	47, 48	syl 17	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = -e -oo )
50		xnegeq 12038	. . . . . . . . 9 |- ( B = -oo -> -e B = -e -oo )
51	50, 42	syl6eq 2672	. . . . . . . 8 |- ( B = -oo -> -e B = +oo )
52	51	oveq2d 6666	. . . . . . 7 |- ( B = -oo -> ( -e A +e -e B ) = ( -e A +e +oo ) )
53		renemnf 10088	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= -oo )
54		xaddpnf1 12057	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( -e A +e +oo ) = +oo )
55	35, 53, 54	syl2anc 693	. . . . . . . 8 |- ( -e A e. RR -> ( -e A +e +oo ) = +oo )
56	34, 55	syl 17	. . . . . . 7 |- ( A e. RR -> ( -e A +e +oo ) = +oo )
57	52, 56	sylan9eqr 2678	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( -e A +e -e B ) = +oo )
58	42, 49, 57	3eqtr4a 2682	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
59	21, 41, 58	3jaodan 1394	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> -e ( A +e B ) = ( -e A +e -e B ) )
60	2, 59	sylan2b 492	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
61		xneg0 12043	. . . . . . 7 |- -e 0 = 0
62		simpr 477	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
63	62	oveq2d 6666	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
64		pnfaddmnf 12061	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
65	63, 64	syl6eq 2672	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = 0 )
66		xnegeq 12038	. . . . . . . 8 |- ( ( +oo +e B ) = 0 -> -e ( +oo +e B ) = -e 0 )
67	65, 66	syl 17	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = -e 0 )
68	51	adantl 482	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> -e B = +oo )
69	68	oveq2d 6666	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = ( -oo +e +oo ) )
70		mnfaddpnf 12062	. . . . . . . 8 |- ( -oo +e +oo ) = 0
71	69, 70	syl6eq 2672	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = 0 )
72	61, 67, 71	3eqtr4a 2682	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
73		xaddpnf2 12058	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
74		xnegeq 12038	. . . . . . . 8 |- ( ( +oo +e B ) = +oo -> -e ( +oo +e B ) = -e +oo )
75	73, 74	syl 17	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = -e +oo )
76		xnegcl 12044	. . . . . . . . 9 |- ( B e. RR* -> -e B e. RR* )
77	76	adantr 481	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B e. RR* )
78		xnegeq 12038	. . . . . . . . . . . 12 |- ( -e B = +oo -> -e -e B = -e +oo )
79	78, 22	syl6eq 2672	. . . . . . . . . . 11 |- ( -e B = +oo -> -e -e B = -oo )
80		xnegneg 12045	. . . . . . . . . . . 12 |- ( B e. RR* -> -e -e B = B )
81	80	eqeq1d 2624	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = -oo <-> B = -oo ) )
82	79, 81	syl5ib 234	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = +oo -> B = -oo ) )
83	82	necon3d 2815	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= -oo -> -e B =/= +oo ) )
84	83	imp 445	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B =/= +oo )
85		xaddmnf2 12060	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= +oo ) -> ( -oo +e -e B ) = -oo )
86	77, 84, 85	syl2anc 693	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( -oo +e -e B ) = -oo )
87	22, 75, 86	3eqtr4a 2682	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
88	72, 87	pm2.61dane 2881	. . . . 5 |- ( B e. RR* -> -e ( +oo +e B ) = ( -oo +e -e B ) )
89	88	adantl 482	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
90		simpl 473	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
91	90	oveq1d 6665	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
92		xnegeq 12038	. . . . 5 |- ( ( A +e B ) = ( +oo +e B ) -> -e ( A +e B ) = -e ( +oo +e B ) )
93	91, 92	syl 17	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> -e ( A +e B ) = -e ( +oo +e B ) )
94		xnegeq 12038	. . . . . . 7 |- ( A = +oo -> -e A = -e +oo )
95	94	adantr 481	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -e +oo )
96	95, 22	syl6eq 2672	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -oo )
97	96	oveq1d 6665	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( -oo +e -e B ) )
98	89, 93, 97	3eqtr4d 2666	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
99		simpr 477	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
100	99	oveq2d 6666	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
101	100, 70	syl6eq 2672	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = 0 )
102		xnegeq 12038	. . . . . . . 8 |- ( ( -oo +e B ) = 0 -> -e ( -oo +e B ) = -e 0 )
103	101, 102	syl 17	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = -e 0 )
104	32	adantl 482	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> -e B = -oo )
105	104	oveq2d 6666	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = ( +oo +e -oo ) )
106	105, 64	syl6eq 2672	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = 0 )
107	61, 103, 106	3eqtr4a 2682	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
108		xaddmnf2 12060	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
109		xnegeq 12038	. . . . . . . 8 |- ( ( -oo +e B ) = -oo -> -e ( -oo +e B ) = -e -oo )
110	108, 109	syl 17	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = -e -oo )
111	76	adantr 481	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B e. RR* )
112		xnegeq 12038	. . . . . . . . . . . 12 |- ( -e B = -oo -> -e -e B = -e -oo )
113	112, 42	syl6eq 2672	. . . . . . . . . . 11 |- ( -e B = -oo -> -e -e B = +oo )
114	80	eqeq1d 2624	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = +oo <-> B = +oo ) )
115	113, 114	syl5ib 234	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = -oo -> B = +oo ) )
116	115	necon3d 2815	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= +oo -> -e B =/= -oo ) )
117	116	imp 445	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B =/= -oo )
118		xaddpnf2 12058	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= -oo ) -> ( +oo +e -e B ) = +oo )
119	111, 117, 118	syl2anc 693	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( +oo +e -e B ) = +oo )
120	42, 110, 119	3eqtr4a 2682	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
121	107, 120	pm2.61dane 2881	. . . . 5 |- ( B e. RR* -> -e ( -oo +e B ) = ( +oo +e -e B ) )
122	121	adantl 482	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
123		simpl 473	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
124	123	oveq1d 6665	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
125		xnegeq 12038	. . . . 5 |- ( ( A +e B ) = ( -oo +e B ) -> -e ( A +e B ) = -e ( -oo +e B ) )
126	124, 125	syl 17	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = -e ( -oo +e B ) )
127		xnegeq 12038	. . . . . . 7 |- ( A = -oo -> -e A = -e -oo )
128	127	adantr 481	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> -e A = -e -oo )
129	128, 42	syl6eq 2672	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> -e A = +oo )
130	129	oveq1d 6665	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( +oo +e -e B ) )
131	122, 126, 130	3eqtr4d 2666	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
132	60, 98, 131	3jaoian 1393	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
133	1, 132	sylanb 489	1 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   -ucneg 10267   -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-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-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:  xaddass2  12080  xposdif  12092  xadddi  12125  xrsxmet  22612