Theorem xnegdi 11485

Description: Extended real version of xnegdi 11485. (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 11367	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 11367	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 9580	. . . . . . . 8 |- ( A e. RR -> A e. CC )
4		recn 9580	. . . . . . . 8 |- ( B e. RR -> B e. CC )
5		negdi 9882	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
6	3, 4, 5	syl2an 479	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u ( A + B ) = ( -u A + -u B ) )
7		readdcl 9573	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexneg 11455	. . . . . . . 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 9888	. . . . . . . 8 |- ( A e. RR -> -u A e. RR )
11		renegcl 9888	. . . . . . . 8 |- ( B e. RR -> -u B e. RR )
12		rexadd 11476	. . . . . . . 8 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
13	10, 11, 12	syl2an 479	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
14	6, 9, 13	3eqtr4d 2472	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = ( -u A +e -u B ) )
15		rexadd 11476	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16		xnegeq 11451	. . . . . . 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 11455	. . . . . . 7 |- ( A e. RR -> -e A = -u A )
19		rexneg 11455	. . . . . . 7 |- ( B e. RR -> -e B = -u B )
20	18, 19	oveqan12d 6268	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -e A +e -e B ) = ( -u A +e -u B ) )
21	14, 17, 20	3eqtr4d 2472	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = ( -e A +e -e B ) )
22		xnegpnf 11453	. . . . . 6 |- -e +oo = -oo
23		oveq2 6257	. . . . . . . 8 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
24		rexr 9637	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
25		renemnf 9640	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
26		xaddpnf1 11470	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
27	24, 25, 26	syl2anc 665	. . . . . . . 8 |- ( A e. RR -> ( A +e +oo ) = +oo )
28	23, 27	sylan9eqr 2484	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
29		xnegeq 11451	. . . . . . 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 11451	. . . . . . . . 9 |- ( B = +oo -> -e B = -e +oo )
32	31, 22	syl6eq 2478	. . . . . . . 8 |- ( B = +oo -> -e B = -oo )
33	32	oveq2d 6265	. . . . . . 7 |- ( B = +oo -> ( -e A +e -e B ) = ( -e A +e -oo ) )
34	18, 10	eqeltrd 2506	. . . . . . . 8 |- ( A e. RR -> -e A e. RR )
35		rexr 9637	. . . . . . . . 9 |- ( -e A e. RR -> -e A e. RR* )
36		renepnf 9639	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= +oo )
37		xaddmnf1 11472	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -e A +e -oo ) = -oo )
38	35, 36, 37	syl2anc 665	. . . . . . . 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 2484	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( -e A +e -e B ) = -oo )
41	22, 30, 40	3eqtr4a 2488	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
42		xnegmnf 11454	. . . . . 6 |- -e -oo = +oo
43		oveq2 6257	. . . . . . . 8 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
44		renepnf 9639	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
45		xaddmnf1 11472	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
46	24, 44, 45	syl2anc 665	. . . . . . . 8 |- ( A e. RR -> ( A +e -oo ) = -oo )
47	43, 46	sylan9eqr 2484	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
48		xnegeq 11451	. . . . . . 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 11451	. . . . . . . . 9 |- ( B = -oo -> -e B = -e -oo )
51	50, 42	syl6eq 2478	. . . . . . . 8 |- ( B = -oo -> -e B = +oo )
52	51	oveq2d 6265	. . . . . . 7 |- ( B = -oo -> ( -e A +e -e B ) = ( -e A +e +oo ) )
53		renemnf 9640	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= -oo )
54		xaddpnf1 11470	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( -e A +e +oo ) = +oo )
55	35, 53, 54	syl2anc 665	. . . . . . . 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 2484	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( -e A +e -e B ) = +oo )
58	42, 49, 57	3eqtr4a 2488	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
59	21, 41, 58	3jaodan 1330	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> -e ( A +e B ) = ( -e A +e -e B ) )
60	2, 59	sylan2b 477	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
61		xneg0 11456	. . . . . . 7 |- -e 0 = 0
62		simpr 462	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
63	62	oveq2d 6265	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
64		pnfaddmnf 11474	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
65	63, 64	syl6eq 2478	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = 0 )
66		xnegeq 11451	. . . . . . . 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 467	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> -e B = +oo )
69	68	oveq2d 6265	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = ( -oo +e +oo ) )
70		mnfaddpnf 11475	. . . . . . . 8 |- ( -oo +e +oo ) = 0
71	69, 70	syl6eq 2478	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = 0 )
72	61, 67, 71	3eqtr4a 2488	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
73		xaddpnf2 11471	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
74		xnegeq 11451	. . . . . . . 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 11457	. . . . . . . . 9 |- ( B e. RR* -> -e B e. RR* )
77	76	adantr 466	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B e. RR* )
78		xnegeq 11451	. . . . . . . . . . . 12 |- ( -e B = +oo -> -e -e B = -e +oo )
79	78, 22	syl6eq 2478	. . . . . . . . . . 11 |- ( -e B = +oo -> -e -e B = -oo )
80		xnegneg 11458	. . . . . . . . . . . 12 |- ( B e. RR* -> -e -e B = B )
81	80	eqeq1d 2430	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = -oo <-> B = -oo ) )
82	79, 81	syl5ib 222	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = +oo -> B = -oo ) )
83	82	necon3d 2622	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= -oo -> -e B =/= +oo ) )
84	83	imp 430	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B =/= +oo )
85		xaddmnf2 11473	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= +oo ) -> ( -oo +e -e B ) = -oo )
86	77, 84, 85	syl2anc 665	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( -oo +e -e B ) = -oo )
87	22, 75, 86	3eqtr4a 2488	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
88	72, 87	pm2.61dane 2688	. . . . 5 |- ( B e. RR* -> -e ( +oo +e B ) = ( -oo +e -e B ) )
89	88	adantl 467	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
90		simpl 458	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
91	90	oveq1d 6264	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
92		xnegeq 11451	. . . . 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 11451	. . . . . . 7 |- ( A = +oo -> -e A = -e +oo )
95	94	adantr 466	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -e +oo )
96	95, 22	syl6eq 2478	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -oo )
97	96	oveq1d 6264	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( -oo +e -e B ) )
98	89, 93, 97	3eqtr4d 2472	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
99		simpr 462	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
100	99	oveq2d 6265	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
101	100, 70	syl6eq 2478	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = 0 )
102		xnegeq 11451	. . . . . . . 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 467	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> -e B = -oo )
105	104	oveq2d 6265	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = ( +oo +e -oo ) )
106	105, 64	syl6eq 2478	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = 0 )
107	61, 103, 106	3eqtr4a 2488	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
108		xaddmnf2 11473	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
109		xnegeq 11451	. . . . . . . 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 466	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B e. RR* )
112		xnegeq 11451	. . . . . . . . . . . 12 |- ( -e B = -oo -> -e -e B = -e -oo )
113	112, 42	syl6eq 2478	. . . . . . . . . . 11 |- ( -e B = -oo -> -e -e B = +oo )
114	80	eqeq1d 2430	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = +oo <-> B = +oo ) )
115	113, 114	syl5ib 222	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = -oo -> B = +oo ) )
116	115	necon3d 2622	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= +oo -> -e B =/= -oo ) )
117	116	imp 430	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B =/= -oo )
118		xaddpnf2 11471	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= -oo ) -> ( +oo +e -e B ) = +oo )
119	111, 117, 118	syl2anc 665	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( +oo +e -e B ) = +oo )
120	42, 110, 119	3eqtr4a 2488	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
121	107, 120	pm2.61dane 2688	. . . . 5 |- ( B e. RR* -> -e ( -oo +e B ) = ( +oo +e -e B ) )
122	121	adantl 467	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
123		simpl 458	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
124	123	oveq1d 6264	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
125		xnegeq 11451	. . . . 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 11451	. . . . . . 7 |- ( A = -oo -> -e A = -e -oo )
128	127	adantr 466	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> -e A = -e -oo )
129	128, 42	syl6eq 2478	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> -e A = +oo )
130	129	oveq1d 6264	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( +oo +e -e B ) )
131	122, 126, 130	3eqtr4d 2472	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
132	60, 98, 131	3jaoian 1329	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
133	1, 132	sylanb 474	1 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )

Syntax hints:   -> wi 4   /\ wa 370   \/ w3o 981   = wceq 1437   e. wcel 1872   =/= wne 2599  (class class class)co 6249   CCcc 9488   RRcr 9489   0cc0 9490   + caddc 9493   +oocpnf 9623   -oocmnf 9624   RR*cxr 9625   -ucneg 9812   -ecxne 11357   +ecxad 11358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-sub 9813  df-neg 9814  df-xneg 11360  df-xadd 11361

This theorem is referenced by:  xaddass2  11487  xposdif  11499  xadddi  11532  xrsxmet  21769