Theorem xnegdi 10025

Description: Extended real version of negdi 8364. (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 9933	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9933	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 8093	. . . . . . . 8 |- ( A e. RR -> A e. CC )
4		recn 8093	. . . . . . . 8 |- ( B e. RR -> B e. CC )
5		negdi 8364	. . . . . . . 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 8086	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexneg 9987	. . . . . . . 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 8368	. . . . . . . 8 |- ( A e. RR -> -u A e. RR )
11		renegcl 8368	. . . . . . . 8 |- ( B e. RR -> -u B e. RR )
12		rexadd 10009	. . . . . . . 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 2250	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = ( -u A +e -u B ) )
15		rexadd 10009	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16		xnegeq 9984	. . . . . . 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 9987	. . . . . . 7 |- ( A e. RR -> -e A = -u A )
19		rexneg 9987	. . . . . . 7 |- ( B e. RR -> -e B = -u B )
20	18, 19	oveqan12d 5986	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -e A +e -e B ) = ( -u A +e -u B ) )
21	14, 17, 20	3eqtr4d 2250	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = ( -e A +e -e B ) )
22		xnegpnf 9985	. . . . . 6 |- -e +oo = -oo
23		oveq2 5975	. . . . . . . 8 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
24		rexr 8153	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
25		renemnf 8156	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
26		xaddpnf1 10003	. . . . . . . . 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 2262	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
29		xnegeq 9984	. . . . . . 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 9984	. . . . . . . . 9 |- ( B = +oo -> -e B = -e +oo )
32	31, 22	eqtrdi 2256	. . . . . . . 8 |- ( B = +oo -> -e B = -oo )
33	32	oveq2d 5983	. . . . . . 7 |- ( B = +oo -> ( -e A +e -e B ) = ( -e A +e -oo ) )
34	18, 10	eqeltrd 2284	. . . . . . . 8 |- ( A e. RR -> -e A e. RR )
35		rexr 8153	. . . . . . . . 9 |- ( -e A e. RR -> -e A e. RR* )
36		renepnf 8155	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= +oo )
37		xaddmnf1 10005	. . . . . . . . 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 2262	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( -e A +e -e B ) = -oo )
41	22, 30, 40	3eqtr4a 2266	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
42		xnegmnf 9986	. . . . . 6 |- -e -oo = +oo
43		oveq2 5975	. . . . . . . 8 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
44		renepnf 8155	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
45		xaddmnf1 10005	. . . . . . . . 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 2262	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
48		xnegeq 9984	. . . . . . 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 9984	. . . . . . . . 9 |- ( B = -oo -> -e B = -e -oo )
51	50, 42	eqtrdi 2256	. . . . . . . 8 |- ( B = -oo -> -e B = +oo )
52	51	oveq2d 5983	. . . . . . 7 |- ( B = -oo -> ( -e A +e -e B ) = ( -e A +e +oo ) )
53		renemnf 8156	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= -oo )
54		xaddpnf1 10003	. . . . . . . . 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 2262	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( -e A +e -e B ) = +oo )
58	42, 49, 57	3eqtr4a 2266	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
59	21, 41, 58	3jaodan 1319	. . . 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 9988	. . . . . . 7 |- -e 0 = 0
62		simpr 110	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
63	62	oveq2d 5983	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
64		pnfaddmnf 10007	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
65	63, 64	eqtrdi 2256	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = 0 )
66		xnegeq 9984	. . . . . . . 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 5983	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = ( -oo +e +oo ) )
70		mnfaddpnf 10008	. . . . . . . 8 |- ( -oo +e +oo ) = 0
71	69, 70	eqtrdi 2256	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = 0 )
72	61, 67, 71	3eqtr4a 2266	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
73		xaddpnf2 10004	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
74		xnegeq 9984	. . . . . . . 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 9989	. . . . . . . 8 |- ( B e. RR* -> -e B e. RR* )
77		xnegeq 9984	. . . . . . . . . . . 12 |- ( -e B = +oo -> -e -e B = -e +oo )
78	77, 22	eqtrdi 2256	. . . . . . . . . . 11 |- ( -e B = +oo -> -e -e B = -oo )
79		xnegneg 9990	. . . . . . . . . . . 12 |- ( B e. RR* -> -e -e B = B )
80	79	eqeq1d 2216	. . . . . . . . . . 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 2422	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= -oo -> -e B =/= +oo ) )
83	82	imp 124	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B =/= +oo )
84		xaddmnf2 10006	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= +oo ) -> ( -oo +e -e B ) = -oo )
85	76, 83, 84	syl2an2r 595	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( -oo +e -e B ) = -oo )
86	22, 75, 85	3eqtr4a 2266	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
87		xrmnfdc 10000	. . . . . . . 8 |- ( B e. RR* -> DECID B = -oo )
88		exmiddc 838	. . . . . . . 8 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
89	87, 88	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
90		df-ne 2379	. . . . . . . 8 |- ( B =/= -oo <-> -. B = -oo )
91	90	orbi2i 764	. . . . . . 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 800	. . . . 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 5982	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
97		xnegeq 9984	. . . . 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 9984	. . . . . . 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 2256	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -oo )
102	101	oveq1d 5982	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( -oo +e -e B ) )
103	94, 98, 102	3eqtr4d 2250	. . 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 5983	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
106	105, 70	eqtrdi 2256	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = 0 )
107		xnegeq 9984	. . . . . . . 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 5983	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = ( +oo +e -oo ) )
111	110, 64	eqtrdi 2256	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = 0 )
112	61, 108, 111	3eqtr4a 2266	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
113		xaddmnf2 10006	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
114		xnegeq 9984	. . . . . . . 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 9984	. . . . . . . . . . . 12 |- ( -e B = -oo -> -e -e B = -e -oo )
117	116, 42	eqtrdi 2256	. . . . . . . . . . 11 |- ( -e B = -oo -> -e -e B = +oo )
118	79	eqeq1d 2216	. . . . . . . . . . 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 2422	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= +oo -> -e B =/= -oo ) )
121	120	imp 124	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B =/= -oo )
122		xaddpnf2 10004	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= -oo ) -> ( +oo +e -e B ) = +oo )
123	76, 121, 122	syl2an2r 595	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( +oo +e -e B ) = +oo )
124	42, 115, 123	3eqtr4a 2266	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
125		xrpnfdc 9999	. . . . . . . 8 |- ( B e. RR* -> DECID B = +oo )
126		exmiddc 838	. . . . . . . 8 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
127	125, 126	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
128		df-ne 2379	. . . . . . . 8 |- ( B =/= +oo <-> -. B = +oo )
129	128	orbi2i 764	. . . . . . 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 800	. . . . 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 5982	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
135		xnegeq 9984	. . . . 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 9984	. . . . . . 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 2256	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> -e A = +oo )
140	139	oveq1d 5982	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( +oo +e -e B ) )
141	132, 136, 140	3eqtr4d 2250	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
142	60, 103, 141	3jaoian 1318	. 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 710  DECID wdc 836   \/ w3o 980   = wceq 1373   e. wcel 2178   =/= wne 2378  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   + caddc 7963   +oocpnf 8139   -oocmnf 8140   RR*cxr 8141   -ucneg 8279   -ecxne 9926   +ecxad 9927

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-sub 8280  df-neg 8281  df-xneg 9929  df-xadd 9930

This theorem is referenced by:  xaddass2  10027  xrminadd  11701