Theorem xnegdi 10783

Description: Extended real version of xnegdi 10783. (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 10672	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 10672	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 9036	. . . . . . . 8 |- ( A e. RR -> A e. CC )
4		recn 9036	. . . . . . . 8 |- ( B e. RR -> B e. CC )
5		negdi 9314	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
6	3, 4, 5	syl2an 464	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u ( A + B ) = ( -u A + -u B ) )
7		readdcl 9029	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexneg 10753	. . . . . . . 8 |- ( ( A + B ) e. RR -> - e ( A + B ) = -u ( A + B ) )
9	7, 8	syl 16	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> - e ( A + B ) = -u ( A + B ) )
10		renegcl 9320	. . . . . . . 8 |- ( A e. RR -> -u A e. RR )
11		renegcl 9320	. . . . . . . 8 |- ( B e. RR -> -u B e. RR )
12		rexadd 10774	. . . . . . . 8 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( -u A + e -u B ) = ( -u A + -u B ) )
13	10, 11, 12	syl2an 464	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A + e -u B ) = ( -u A + -u B ) )
14	6, 9, 13	3eqtr4d 2446	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> - e ( A + B ) = ( -u A + e -u B ) )
15		rexadd 10774	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + e B ) = ( A + B ) )
16		xnegeq 10749	. . . . . . 7 |- ( ( A + e B ) = ( A + B ) -> - e ( A + e B ) = - e ( A + B ) )
17	15, 16	syl 16	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> - e ( A + e B ) = - e ( A + B ) )
18		rexneg 10753	. . . . . . 7 |- ( A e. RR -> - e A = -u A )
19		rexneg 10753	. . . . . . 7 |- ( B e. RR -> - e B = -u B )
20	18, 19	oveqan12d 6059	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( - e A + e - e B ) = ( -u A + e -u B ) )
21	14, 17, 20	3eqtr4d 2446	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> - e ( A + e B ) = ( - e A + e - e B ) )
22		xnegpnf 10751	. . . . . 6 |- - e +oo = -oo
23		oveq2 6048	. . . . . . . 8 |- ( B = +oo -> ( A + e B ) = ( A + e +oo ) )
24		rexr 9086	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
25		renemnf 9089	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
26		xaddpnf1 10768	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A + e +oo ) = +oo )
27	24, 25, 26	syl2anc 643	. . . . . . . 8 |- ( A e. RR -> ( A + e +oo ) = +oo )
28	23, 27	sylan9eqr 2458	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A + e B ) = +oo )
29		xnegeq 10749	. . . . . . 7 |- ( ( A + e B ) = +oo -> - e ( A + e B ) = - e +oo )
30	28, 29	syl 16	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> - e ( A + e B ) = - e +oo )
31		xnegeq 10749	. . . . . . . . 9 |- ( B = +oo -> - e B = - e +oo )
32	31, 22	syl6eq 2452	. . . . . . . 8 |- ( B = +oo -> - e B = -oo )
33	32	oveq2d 6056	. . . . . . 7 |- ( B = +oo -> ( - e A + e - e B ) = ( - e A + e -oo ) )
34	18, 10	eqeltrd 2478	. . . . . . . 8 |- ( A e. RR -> - e A e. RR )
35		rexr 9086	. . . . . . . . 9 |- ( - e A e. RR -> - e A e. RR* )
36		renepnf 9088	. . . . . . . . 9 |- ( - e A e. RR -> - e A =/= +oo )
37		xaddmnf1 10770	. . . . . . . . 9 |- ( ( - e A e. RR* /\ - e A =/= +oo ) -> ( - e A + e -oo ) = -oo )
38	35, 36, 37	syl2anc 643	. . . . . . . 8 |- ( - e A e. RR -> ( - e A + e -oo ) = -oo )
39	34, 38	syl 16	. . . . . . 7 |- ( A e. RR -> ( - e A + e -oo ) = -oo )
40	33, 39	sylan9eqr 2458	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( - e A + e - e B ) = -oo )
41	22, 30, 40	3eqtr4a 2462	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> - e ( A + e B ) = ( - e A + e - e B ) )
42		xnegmnf 10752	. . . . . 6 |- - e -oo = +oo
43		oveq2 6048	. . . . . . . 8 |- ( B = -oo -> ( A + e B ) = ( A + e -oo ) )
44		renepnf 9088	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
45		xaddmnf1 10770	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A + e -oo ) = -oo )
46	24, 44, 45	syl2anc 643	. . . . . . . 8 |- ( A e. RR -> ( A + e -oo ) = -oo )
47	43, 46	sylan9eqr 2458	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A + e B ) = -oo )
48		xnegeq 10749	. . . . . . 7 |- ( ( A + e B ) = -oo -> - e ( A + e B ) = - e -oo )
49	47, 48	syl 16	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> - e ( A + e B ) = - e -oo )
50		xnegeq 10749	. . . . . . . . 9 |- ( B = -oo -> - e B = - e -oo )
51	50, 42	syl6eq 2452	. . . . . . . 8 |- ( B = -oo -> - e B = +oo )
52	51	oveq2d 6056	. . . . . . 7 |- ( B = -oo -> ( - e A + e - e B ) = ( - e A + e +oo ) )
53		renemnf 9089	. . . . . . . . 9 |- ( - e A e. RR -> - e A =/= -oo )
54		xaddpnf1 10768	. . . . . . . . 9 |- ( ( - e A e. RR* /\ - e A =/= -oo ) -> ( - e A + e +oo ) = +oo )
55	35, 53, 54	syl2anc 643	. . . . . . . 8 |- ( - e A e. RR -> ( - e A + e +oo ) = +oo )
56	34, 55	syl 16	. . . . . . 7 |- ( A e. RR -> ( - e A + e +oo ) = +oo )
57	52, 56	sylan9eqr 2458	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( - e A + e - e B ) = +oo )
58	42, 49, 57	3eqtr4a 2462	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> - e ( A + e B ) = ( - e A + e - e B ) )
59	21, 41, 58	3jaodan 1250	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> - e ( A + e B ) = ( - e A + e - e B ) )
60	2, 59	sylan2b 462	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> - e ( A + e B ) = ( - e A + e - e B ) )
61		xneg0 10754	. . . . . . 7 |- - e 0 = 0
62		simpr 448	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
63	62	oveq2d 6056	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo + e B ) = ( +oo + e -oo ) )
64		pnfaddmnf 10772	. . . . . . . . 9 |- ( +oo + e -oo ) = 0
65	63, 64	syl6eq 2452	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo + e B ) = 0 )
66		xnegeq 10749	. . . . . . . 8 |- ( ( +oo + e B ) = 0 -> - e ( +oo + e B ) = - e 0 )
67	65, 66	syl 16	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> - e ( +oo + e B ) = - e 0 )
68	51	adantl 453	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> - e B = +oo )
69	68	oveq2d 6056	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo + e - e B ) = ( -oo + e +oo ) )
70		mnfaddpnf 10773	. . . . . . . 8 |- ( -oo + e +oo ) = 0
71	69, 70	syl6eq 2452	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo + e - e B ) = 0 )
72	61, 67, 71	3eqtr4a 2462	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> - e ( +oo + e B ) = ( -oo + e - e B ) )
73		xaddpnf2 10769	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo + e B ) = +oo )
74		xnegeq 10749	. . . . . . . 8 |- ( ( +oo + e B ) = +oo -> - e ( +oo + e B ) = - e +oo )
75	73, 74	syl 16	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> - e ( +oo + e B ) = - e +oo )
76		xnegcl 10755	. . . . . . . . 9 |- ( B e. RR* -> - e B e. RR* )
77	76	adantr 452	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> - e B e. RR* )
78		xnegeq 10749	. . . . . . . . . . . 12 |- ( - e B = +oo -> - e - e B = - e +oo )
79	78, 22	syl6eq 2452	. . . . . . . . . . 11 |- ( - e B = +oo -> - e - e B = -oo )
80		xnegneg 10756	. . . . . . . . . . . 12 |- ( B e. RR* -> - e - e B = B )
81	80	eqeq1d 2412	. . . . . . . . . . 11 |- ( B e. RR* -> ( - e - e B = -oo <-> B = -oo ) )
82	79, 81	syl5ib 211	. . . . . . . . . 10 |- ( B e. RR* -> ( - e B = +oo -> B = -oo ) )
83	82	necon3d 2605	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= -oo -> - e B =/= +oo ) )
84	83	imp 419	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> - e B =/= +oo )
85		xaddmnf2 10771	. . . . . . . 8 |- ( ( - e B e. RR* /\ - e B =/= +oo ) -> ( -oo + e - e B ) = -oo )
86	77, 84, 85	syl2anc 643	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( -oo + e - e B ) = -oo )
87	22, 75, 86	3eqtr4a 2462	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> - e ( +oo + e B ) = ( -oo + e - e B ) )
88	72, 87	pm2.61dane 2645	. . . . 5 |- ( B e. RR* -> - e ( +oo + e B ) = ( -oo + e - e B ) )
89	88	adantl 453	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> - e ( +oo + e B ) = ( -oo + e - e B ) )
90		simpl 444	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
91	90	oveq1d 6055	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A + e B ) = ( +oo + e B ) )
92		xnegeq 10749	. . . . 5 |- ( ( A + e B ) = ( +oo + e B ) -> - e ( A + e B ) = - e ( +oo + e B ) )
93	91, 92	syl 16	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> - e ( A + e B ) = - e ( +oo + e B ) )
94		xnegeq 10749	. . . . . . 7 |- ( A = +oo -> - e A = - e +oo )
95	94	adantr 452	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> - e A = - e +oo )
96	95, 22	syl6eq 2452	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> - e A = -oo )
97	96	oveq1d 6055	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( - e A + e - e B ) = ( -oo + e - e B ) )
98	89, 93, 97	3eqtr4d 2446	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> - e ( A + e B ) = ( - e A + e - e B ) )
99		simpr 448	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
100	99	oveq2d 6056	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo + e B ) = ( -oo + e +oo ) )
101	100, 70	syl6eq 2452	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo + e B ) = 0 )
102		xnegeq 10749	. . . . . . . 8 |- ( ( -oo + e B ) = 0 -> - e ( -oo + e B ) = - e 0 )
103	101, 102	syl 16	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> - e ( -oo + e B ) = - e 0 )
104	32	adantl 453	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> - e B = -oo )
105	104	oveq2d 6056	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo + e - e B ) = ( +oo + e -oo ) )
106	105, 64	syl6eq 2452	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo + e - e B ) = 0 )
107	61, 103, 106	3eqtr4a 2462	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> - e ( -oo + e B ) = ( +oo + e - e B ) )
108		xaddmnf2 10771	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo + e B ) = -oo )
109		xnegeq 10749	. . . . . . . 8 |- ( ( -oo + e B ) = -oo -> - e ( -oo + e B ) = - e -oo )
110	108, 109	syl 16	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> - e ( -oo + e B ) = - e -oo )
111	76	adantr 452	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> - e B e. RR* )
112		xnegeq 10749	. . . . . . . . . . . 12 |- ( - e B = -oo -> - e - e B = - e -oo )
113	112, 42	syl6eq 2452	. . . . . . . . . . 11 |- ( - e B = -oo -> - e - e B = +oo )
114	80	eqeq1d 2412	. . . . . . . . . . 11 |- ( B e. RR* -> ( - e - e B = +oo <-> B = +oo ) )
115	113, 114	syl5ib 211	. . . . . . . . . 10 |- ( B e. RR* -> ( - e B = -oo -> B = +oo ) )
116	115	necon3d 2605	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= +oo -> - e B =/= -oo ) )
117	116	imp 419	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> - e B =/= -oo )
118		xaddpnf2 10769	. . . . . . . 8 |- ( ( - e B e. RR* /\ - e B =/= -oo ) -> ( +oo + e - e B ) = +oo )
119	111, 117, 118	syl2anc 643	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( +oo + e - e B ) = +oo )
120	42, 110, 119	3eqtr4a 2462	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> - e ( -oo + e B ) = ( +oo + e - e B ) )
121	107, 120	pm2.61dane 2645	. . . . 5 |- ( B e. RR* -> - e ( -oo + e B ) = ( +oo + e - e B ) )
122	121	adantl 453	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> - e ( -oo + e B ) = ( +oo + e - e B ) )
123		simpl 444	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
124	123	oveq1d 6055	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> ( A + e B ) = ( -oo + e B ) )
125		xnegeq 10749	. . . . 5 |- ( ( A + e B ) = ( -oo + e B ) -> - e ( A + e B ) = - e ( -oo + e B ) )
126	124, 125	syl 16	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> - e ( A + e B ) = - e ( -oo + e B ) )
127		xnegeq 10749	. . . . . . 7 |- ( A = -oo -> - e A = - e -oo )
128	127	adantr 452	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> - e A = - e -oo )
129	128, 42	syl6eq 2452	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> - e A = +oo )
130	129	oveq1d 6055	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( - e A + e - e B ) = ( +oo + e - e B ) )
131	122, 126, 130	3eqtr4d 2446	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> - e ( A + e B ) = ( - e A + e - e B ) )
132	60, 98, 131	3jaoian 1249	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> - e ( A + e B ) = ( - e A + e - e B ) )
133	1, 132	sylanb 459	1 |- ( ( A e. RR* /\ B e. RR* ) -> - e ( A + e B ) = ( - e A + e - e B ) )

Syntax hints:   -> wi 4   /\ wa 359   \/ w3o 935   = wceq 1649   e. wcel 1721   =/= wne 2567  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   + caddc 8949   +oocpnf 9073   -oocmnf 9074   RR*cxr 9075   -ucneg 9248   - ecxne 10663   + ecxad 10664

This theorem is referenced by:  xaddass2  10785  xposdif  10797  xadddi  10830  xrsxmet  18793

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-sub 9249  df-neg 9250  df-xneg 10666  df-xadd 10667