Theorem xnegdi 11431

Description: Extended real version of xnegdi 11431. (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 11316	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 11316	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 9573	. . . . . . . 8 |- ( A e. RR -> A e. CC )
4		recn 9573	. . . . . . . 8 |- ( B e. RR -> B e. CC )
5		negdi 9867	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
6	3, 4, 5	syl2an 477	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> -u ( A + B ) = ( -u A + -u B ) )
7		readdcl 9566	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexneg 11401	. . . . . . . 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 9873	. . . . . . . 8 |- ( A e. RR -> -u A e. RR )
11		renegcl 9873	. . . . . . . 8 |- ( B e. RR -> -u B e. RR )
12		rexadd 11422	. . . . . . . 8 |- ( ( -u A e. RR /\ -u B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
13	10, 11, 12	syl2an 477	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -u A +e -u B ) = ( -u A + -u B ) )
14	6, 9, 13	3eqtr4d 2513	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -e ( A + B ) = ( -u A +e -u B ) )
15		rexadd 11422	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16		xnegeq 11397	. . . . . . 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 11401	. . . . . . 7 |- ( A e. RR -> -e A = -u A )
19		rexneg 11401	. . . . . . 7 |- ( B e. RR -> -e B = -u B )
20	18, 19	oveqan12d 6296	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -e A +e -e B ) = ( -u A +e -u B ) )
21	14, 17, 20	3eqtr4d 2513	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> -e ( A +e B ) = ( -e A +e -e B ) )
22		xnegpnf 11399	. . . . . 6 |- -e +oo = -oo
23		oveq2 6285	. . . . . . . 8 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
24		rexr 9630	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
25		renemnf 9633	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
26		xaddpnf1 11416	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
27	24, 25, 26	syl2anc 661	. . . . . . . 8 |- ( A e. RR -> ( A +e +oo ) = +oo )
28	23, 27	sylan9eqr 2525	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
29		xnegeq 11397	. . . . . . 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 11397	. . . . . . . . 9 |- ( B = +oo -> -e B = -e +oo )
32	31, 22	syl6eq 2519	. . . . . . . 8 |- ( B = +oo -> -e B = -oo )
33	32	oveq2d 6293	. . . . . . 7 |- ( B = +oo -> ( -e A +e -e B ) = ( -e A +e -oo ) )
34	18, 10	eqeltrd 2550	. . . . . . . 8 |- ( A e. RR -> -e A e. RR )
35		rexr 9630	. . . . . . . . 9 |- ( -e A e. RR -> -e A e. RR* )
36		renepnf 9632	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= +oo )
37		xaddmnf1 11418	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= +oo ) -> ( -e A +e -oo ) = -oo )
38	35, 36, 37	syl2anc 661	. . . . . . . 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 2525	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( -e A +e -e B ) = -oo )
41	22, 30, 40	3eqtr4a 2529	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
42		xnegmnf 11400	. . . . . 6 |- -e -oo = +oo
43		oveq2 6285	. . . . . . . 8 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
44		renepnf 9632	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
45		xaddmnf1 11418	. . . . . . . . 9 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
46	24, 44, 45	syl2anc 661	. . . . . . . 8 |- ( A e. RR -> ( A +e -oo ) = -oo )
47	43, 46	sylan9eqr 2525	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
48		xnegeq 11397	. . . . . . 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 11397	. . . . . . . . 9 |- ( B = -oo -> -e B = -e -oo )
51	50, 42	syl6eq 2519	. . . . . . . 8 |- ( B = -oo -> -e B = +oo )
52	51	oveq2d 6293	. . . . . . 7 |- ( B = -oo -> ( -e A +e -e B ) = ( -e A +e +oo ) )
53		renemnf 9633	. . . . . . . . 9 |- ( -e A e. RR -> -e A =/= -oo )
54		xaddpnf1 11416	. . . . . . . . 9 |- ( ( -e A e. RR* /\ -e A =/= -oo ) -> ( -e A +e +oo ) = +oo )
55	35, 53, 54	syl2anc 661	. . . . . . . 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 2525	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( -e A +e -e B ) = +oo )
58	42, 49, 57	3eqtr4a 2529	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> -e ( A +e B ) = ( -e A +e -e B ) )
59	21, 41, 58	3jaodan 1289	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> -e ( A +e B ) = ( -e A +e -e B ) )
60	2, 59	sylan2b 475	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
61		xneg0 11402	. . . . . . 7 |- -e 0 = 0
62		simpr 461	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
63	62	oveq2d 6293	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
64		pnfaddmnf 11420	. . . . . . . . 9 |- ( +oo +e -oo ) = 0
65	63, 64	syl6eq 2519	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = 0 )
66		xnegeq 11397	. . . . . . . 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 466	. . . . . . . . 9 |- ( ( B e. RR* /\ B = -oo ) -> -e B = +oo )
69	68	oveq2d 6293	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = ( -oo +e +oo ) )
70		mnfaddpnf 11421	. . . . . . . 8 |- ( -oo +e +oo ) = 0
71	69, 70	syl6eq 2519	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( -oo +e -e B ) = 0 )
72	61, 67, 71	3eqtr4a 2529	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
73		xaddpnf2 11417	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
74		xnegeq 11397	. . . . . . . 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 11403	. . . . . . . . 9 |- ( B e. RR* -> -e B e. RR* )
77	76	adantr 465	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B e. RR* )
78		xnegeq 11397	. . . . . . . . . . . 12 |- ( -e B = +oo -> -e -e B = -e +oo )
79	78, 22	syl6eq 2519	. . . . . . . . . . 11 |- ( -e B = +oo -> -e -e B = -oo )
80		xnegneg 11404	. . . . . . . . . . . 12 |- ( B e. RR* -> -e -e B = B )
81	80	eqeq1d 2464	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = -oo <-> B = -oo ) )
82	79, 81	syl5ib 219	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = +oo -> B = -oo ) )
83	82	necon3d 2686	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= -oo -> -e B =/= +oo ) )
84	83	imp 429	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= -oo ) -> -e B =/= +oo )
85		xaddmnf2 11419	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= +oo ) -> ( -oo +e -e B ) = -oo )
86	77, 84, 85	syl2anc 661	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( -oo +e -e B ) = -oo )
87	22, 75, 86	3eqtr4a 2529	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
88	72, 87	pm2.61dane 2780	. . . . 5 |- ( B e. RR* -> -e ( +oo +e B ) = ( -oo +e -e B ) )
89	88	adantl 466	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> -e ( +oo +e B ) = ( -oo +e -e B ) )
90		simpl 457	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
91	90	oveq1d 6292	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
92		xnegeq 11397	. . . . 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 11397	. . . . . . 7 |- ( A = +oo -> -e A = -e +oo )
95	94	adantr 465	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -e +oo )
96	95, 22	syl6eq 2519	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> -e A = -oo )
97	96	oveq1d 6292	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( -oo +e -e B ) )
98	89, 93, 97	3eqtr4d 2513	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
99		simpr 461	. . . . . . . . . 10 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
100	99	oveq2d 6293	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
101	100, 70	syl6eq 2519	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = 0 )
102		xnegeq 11397	. . . . . . . 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 466	. . . . . . . . 9 |- ( ( B e. RR* /\ B = +oo ) -> -e B = -oo )
105	104	oveq2d 6293	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = ( +oo +e -oo ) )
106	105, 64	syl6eq 2519	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( +oo +e -e B ) = 0 )
107	61, 103, 106	3eqtr4a 2529	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
108		xaddmnf2 11419	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
109		xnegeq 11397	. . . . . . . 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 465	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B e. RR* )
112		xnegeq 11397	. . . . . . . . . . . 12 |- ( -e B = -oo -> -e -e B = -e -oo )
113	112, 42	syl6eq 2519	. . . . . . . . . . 11 |- ( -e B = -oo -> -e -e B = +oo )
114	80	eqeq1d 2464	. . . . . . . . . . 11 |- ( B e. RR* -> ( -e -e B = +oo <-> B = +oo ) )
115	113, 114	syl5ib 219	. . . . . . . . . 10 |- ( B e. RR* -> ( -e B = -oo -> B = +oo ) )
116	115	necon3d 2686	. . . . . . . . 9 |- ( B e. RR* -> ( B =/= +oo -> -e B =/= -oo ) )
117	116	imp 429	. . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> -e B =/= -oo )
118		xaddpnf2 11417	. . . . . . . 8 |- ( ( -e B e. RR* /\ -e B =/= -oo ) -> ( +oo +e -e B ) = +oo )
119	111, 117, 118	syl2anc 661	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( +oo +e -e B ) = +oo )
120	42, 110, 119	3eqtr4a 2529	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
121	107, 120	pm2.61dane 2780	. . . . 5 |- ( B e. RR* -> -e ( -oo +e B ) = ( +oo +e -e B ) )
122	121	adantl 466	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> -e ( -oo +e B ) = ( +oo +e -e B ) )
123		simpl 457	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
124	123	oveq1d 6292	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
125		xnegeq 11397	. . . . 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 11397	. . . . . . 7 |- ( A = -oo -> -e A = -e -oo )
128	127	adantr 465	. . . . . 6 |- ( ( A = -oo /\ B e. RR* ) -> -e A = -e -oo )
129	128, 42	syl6eq 2519	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> -e A = +oo )
130	129	oveq1d 6292	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -e A +e -e B ) = ( +oo +e -e B ) )
131	122, 126, 130	3eqtr4d 2513	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
132	60, 98, 131	3jaoian 1288	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
133	1, 132	sylanb 472	1 |- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )

Syntax hints:   -> wi 4   /\ wa 369   \/ w3o 967   = wceq 1374   e. wcel 1762   =/= wne 2657  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   + caddc 9486   +oocpnf 9616   -oocmnf 9617   RR*cxr 9618   -ucneg 9797   -ecxne 11306   +ecxad 11307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-sub 9798  df-neg 9799  df-xneg 11309  df-xadd 11310

This theorem is referenced by:  xaddass2  11433  xposdif  11445  xadddi  11478  xrsxmet  21044