Theorem xnegdi 10201

Description: Extended real version of negdi 8530. (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 10109	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 10109	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 8260	. . . . . . . 8 |- ( A e. RR -> A e. CC )
4		recn 8260	. . . . . . . 8 |- ( B e. RR -> B e. CC )
5		negdi 8530	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
6	3, 4, 5	syl2an 289	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u ( A + B ) = ( -u A + -u B ) )
7		readdcl 8253	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexneg 10163	. . . . . . . 8 |- ( ( A + B ) e. RR -> -e ( A + B ) = -u ( A + B ) )
9	7, 8	syl 14	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = -u ( A + B ) )
10		renegcl 8534	. . . . . . . 8 |- ( A e. RR -> -u A e. RR )
11		renegcl 8534	. . . . . . . 8 |- ( B e. RR -> -u B e. RR )
12		rexadd 10185	. . . . . . . 8 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
13	10, 11, 12	syl2an 289	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
14	6, 9, 13	3eqtr4d 2275	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = ( -u A +e -u B ) )
15		rexadd 10185	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16		xnegeq 10160	. . . . . . 7 |- ( ( A +e B ) = ( A + B ) -> -e ( A +e B ) = -e ( A + B ) )
17	15, 16	syl 14	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = -e ( A + B ) )
18		rexneg 10163	. . . . . . 7 |- ( A e. RR -> -e A = -u A )
19		rexneg 10163	. . . . . . 7 |- ( B e. RR -> -e B = -u B )
20	18, 19	oveqan12d 6069	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -e A +e -e B ) = ( -u A +e -u B ) )
21	14, 17, 20	3eqtr4d 2275	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = ( -e A +e -e B ) )
22		xnegpnf 10161	. . . . . 6 |- -e +oo = -oo
23		oveq2 6058	. . . . . . . 8 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
24		rexr 8319	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
25		renemnf 8322	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
26		xaddpnf1 10179	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
27	24, 25, 26	syl2anc 411	. . . . . . . 8 |- ( A e. RR -> ( A +e +oo ) = +oo )
28	23, 27	sylan9eqr 2287	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
29		xnegeq 10160	. . . . . . 7 |- ( ( A +e B ) = +oo -> -e ( A +e B ) = -e +oo )
30	28, 29	syl 14	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = -e +oo )
31		xnegeq 10160	. . . . . . . . 9 |- ( B = +oo -> -e B = -e +oo )
32	31, 22	eqtrdi 2281	. . . . . . . 8 |- ( B = +oo -> -e B = -oo )
33	32	oveq2d 6066	. . . . . . 7 |- ( B = +oo -> ( -e A +e -e B ) = ( -e A +e -oo ) )
34	18, 10	eqeltrd 2309	. . . . . . . 8 |- ( A e. RR -> -e A e. RR )
35		rexr 8319	. . . . . . . . 9 |- ( -e A e. RR -> -e A e. RR* )
36		renepnf 8321	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= +oo )
37		xaddmnf1 10181	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -e A +e -oo ) = -oo )
38	35, 36, 37	syl2anc 411	. . . . . . . 8 |- ( -e A e. RR -> ( -e A +e -oo ) = -oo )
39	34, 38	syl 14	. . . . . . 7 |- ( A e. RR -> ( -e A +e -oo ) = -oo )
40	33, 39	sylan9eqr 2287	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( -e A +e -e B ) = -oo )
41	22, 30, 40	3eqtr4a 2291	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
42		xnegmnf 10162	. . . . . 6 |- -e -oo = +oo
43		oveq2 6058	. . . . . . . 8 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
44		renepnf 8321	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
45		xaddmnf1 10181	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
46	24, 44, 45	syl2anc 411	. . . . . . . 8 |- ( A e. RR -> ( A +e -oo ) = -oo )
47	43, 46	sylan9eqr 2287	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
48		xnegeq 10160	. . . . . . 7 |- ( ( A +e B ) = -oo -> -e ( A +e B ) = -e -oo )
49	47, 48	syl 14	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = -e -oo )
50		xnegeq 10160	. . . . . . . . 9 |- ( B = -oo -> -e B = -e -oo )
51	50, 42	eqtrdi 2281	. . . . . . . 8 |- ( B = -oo -> -e B = +oo )
52	51	oveq2d 6066	. . . . . . 7 |- ( B = -oo -> ( -e A +e -e B ) = ( -e A +e +oo ) )
53		renemnf 8322	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= -oo )
54		xaddpnf1 10179	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( -e A +e +oo ) = +oo )
55	35, 53, 54	syl2anc 411	. . . . . . . 8 |- ( -e A e. RR -> ( -e A +e +oo ) = +oo )
56	34, 55	syl 14	. . . . . . 7 |- ( A e. RR -> ( -e A +e +oo ) = +oo )
57	52, 56	sylan9eqr 2287	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( -e A +e -e B ) = +oo )
58	42, 49, 57	3eqtr4a 2291	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
59	21, 41, 58	3jaodan 1343	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> -e ( A +e B ) = ( -e A +e -e B ) )
60	2, 59	sylan2b 287	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
61		xneg0 10164	. . . . . . 7 |- -e 0 = 0
62		simpr 110	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
63	62	oveq2d 6066	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
64		pnfaddmnf 10183	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
65	63, 64	eqtrdi 2281	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = 0 )
66		xnegeq 10160	. . . . . . . 8 |- ( ( +oo +e B ) = 0 -> -e ( +oo +e B ) = -e 0 )
67	65, 66	syl 14	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = -e 0 )
68	51	adantl 277	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> -e B = +oo )
69	68	oveq2d 6066	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = ( -oo +e +oo ) )
70		mnfaddpnf 10184	. . . . . . . 8 |- ( -oo +e +oo ) = 0
71	69, 70	eqtrdi 2281	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = 0 )
72	61, 67, 71	3eqtr4a 2291	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
73		xaddpnf2 10180	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
74		xnegeq 10160	. . . . . . . 8 |- ( ( +oo +e B ) = +oo -> -e ( +oo +e B ) = -e +oo )
75	73, 74	syl 14	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = -e +oo )
76		xnegcl 10165	. . . . . . . 8 |- ( B e. RR* -> -e B e. RR* )
77		xnegeq 10160	. . . . . . . . . . . 12 |- ( -e B = +oo -> -e -e B = -e +oo )
78	77, 22	eqtrdi 2281	. . . . . . . . . . 11 |- ( -e B = +oo -> -e -e B = -oo )
79		xnegneg 10166	. . . . . . . . . . . 12 |- ( B e. RR* -> -e -e B = B )
80	79	eqeq1d 2241	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = -oo <-> B = -oo ) )
81	78, 80	imbitrid 154	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = +oo -> B = -oo ) )
82	81	necon3d 2456	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= -oo -> -e B =/= +oo ) )
83	82	imp 124	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B =/= +oo )
84		xaddmnf2 10182	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= +oo ) -> ( -oo +e -e B ) = -oo )
85	76, 83, 84	syl2an2r 599	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( -oo +e -e B ) = -oo )
86	22, 75, 85	3eqtr4a 2291	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
87		xrmnfdc 10176	. . . . . . . 8 |- ( B e. RR* -> DECID B = -oo )
88		exmiddc 844	. . . . . . . 8 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
89	87, 88	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
90		df-ne 2413	. . . . . . . 8 |- ( B =/= -oo <-> -. B = -oo )
91	90	orbi2i 770	. . . . . . 7 |- ( ( B = -oo \/ B =/= -oo ) <-> ( B = -oo \/ -. B = -oo ) )
92	89, 91	sylibr 134	. . . . . 6 |- ( B e. RR* -> ( B = -oo \/ B =/= -oo ) )
93	72, 86, 92	mpjaodan 806	. . . . 5 |- ( B e. RR* -> -e ( +oo +e B ) = ( -oo +e -e B ) )
94	93	adantl 277	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
95		simpl 109	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
96	95	oveq1d 6065	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
97		xnegeq 10160	. . . . 5 |- ( ( A +e B ) = ( +oo +e B ) -> -e ( A +e B ) = -e ( +oo +e B ) )
98	96, 97	syl 14	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> -e ( A +e B ) = -e ( +oo +e B ) )
99		xnegeq 10160	. . . . . . 7 |- ( A = +oo -> -e A = -e +oo )
100	99	adantr 276	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -e +oo )
101	100, 22	eqtrdi 2281	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -oo )
102	101	oveq1d 6065	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( -oo +e -e B ) )
103	94, 98, 102	3eqtr4d 2275	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
104		simpr 110	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
105	104	oveq2d 6066	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
106	105, 70	eqtrdi 2281	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = 0 )
107		xnegeq 10160	. . . . . . . 8 |- ( ( -oo +e B ) = 0 -> -e ( -oo +e B ) = -e 0 )
108	106, 107	syl 14	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = -e 0 )
109	32	adantl 277	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> -e B = -oo )
110	109	oveq2d 6066	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = ( +oo +e -oo ) )
111	110, 64	eqtrdi 2281	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = 0 )
112	61, 108, 111	3eqtr4a 2291	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
113		xaddmnf2 10182	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
114		xnegeq 10160	. . . . . . . 8 |- ( ( -oo +e B ) = -oo -> -e ( -oo +e B ) = -e -oo )
115	113, 114	syl 14	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = -e -oo )
116		xnegeq 10160	. . . . . . . . . . . 12 |- ( -e B = -oo -> -e -e B = -e -oo )
117	116, 42	eqtrdi 2281	. . . . . . . . . . 11 |- ( -e B = -oo -> -e -e B = +oo )
118	79	eqeq1d 2241	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = +oo <-> B = +oo ) )
119	117, 118	imbitrid 154	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = -oo -> B = +oo ) )
120	119	necon3d 2456	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= +oo -> -e B =/= -oo ) )
121	120	imp 124	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B =/= -oo )
122		xaddpnf2 10180	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= -oo ) -> ( +oo +e -e B ) = +oo )
123	76, 121, 122	syl2an2r 599	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( +oo +e -e B ) = +oo )
124	42, 115, 123	3eqtr4a 2291	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
125		xrpnfdc 10175	. . . . . . . 8 |- ( B e. RR* -> DECID B = +oo )
126		exmiddc 844	. . . . . . . 8 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
127	125, 126	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
128		df-ne 2413	. . . . . . . 8 |- ( B =/= +oo <-> -. B = +oo )
129	128	orbi2i 770	. . . . . . 7 |- ( ( B = +oo \/ B =/= +oo ) <-> ( B = +oo \/ -. B = +oo ) )
130	127, 129	sylibr 134	. . . . . 6 |- ( B e. RR* -> ( B = +oo \/ B =/= +oo ) )
131	112, 124, 130	mpjaodan 806	. . . . 5 |- ( B e. RR* -> -e ( -oo +e B ) = ( +oo +e -e B ) )
132	131	adantl 277	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
133		simpl 109	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
134	133	oveq1d 6065	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
135		xnegeq 10160	. . . . 5 |- ( ( A +e B ) = ( -oo +e B ) -> -e ( A +e B ) = -e ( -oo +e B ) )
136	134, 135	syl 14	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = -e ( -oo +e B ) )
137		xnegeq 10160	. . . . . . 7 |- ( A = -oo -> -e A = -e -oo )
138	137	adantr 276	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> -e A = -e -oo )
139	138, 42	eqtrdi 2281	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> -e A = +oo )
140	139	oveq1d 6065	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( +oo +e -e B ) )
141	132, 136, 140	3eqtr4d 2275	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
142	60, 103, 141	3jaoian 1342	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
143	1, 142	sylanb 284	1 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2203   =/= wne 2412  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   + caddc 8130   +oocpnf 8305   -oocmnf 8306   RR*cxr 8307   -ucneg 8445   -ecxne 10102   +ecxad 10103

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-sub 8446  df-neg 8447  df-xneg 10105  df-xadd 10106

This theorem is referenced by:  xaddass2  10203  xrminadd  11960