Theorem xnegneg 9190

Description: Extended real version of negneg 7635. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xnegneg	|- ( A e. RR* -> -e -e A = A )

Proof of Theorem xnegneg

Step	Hyp	Ref	Expression
1		elxr 9142	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9187	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9184	. . . . 5 |- ( -e A = -u A -> -e -e A = -e -u A )
4	2, 3	syl 14	. . . 4 |- ( A e. RR -> -e -e A = -e -u A )
5		renegcl 7646	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9187	. . . . 5 |- ( -u A e. RR -> -e -u A = -u -u A )
7	5, 6	syl 14	. . . 4 |- ( A e. RR -> -e -u A = -u -u A )
8		recn 7378	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 7687	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2119	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9186	. . . 4 |- -e -oo = +oo
12		xnegeq 9184	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9185	. . . . . 6 |- -e +oo = -oo
14	12, 13	syl6eq 2131	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9184	. . . . 5 |- ( -e A = -oo -> -e -e A = -e -oo )
16	14, 15	syl 14	. . . 4 |- ( A = +oo -> -e -e A = -e -oo )
17		id 19	. . . 4 |- ( A = +oo -> A = +oo )
18	11, 16, 17	3eqtr4a 2141	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9184	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	syl6eq 2131	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9184	. . . . 5 |- ( -e A = +oo -> -e -e A = -e +oo )
22	20, 21	syl 14	. . . 4 |- ( A = -oo -> -e -e A = -e +oo )
23		id 19	. . . 4 |- ( A = -oo -> A = -oo )
24	13, 22, 23	3eqtr4a 2141	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1235	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> -e -e A = A )
26	1, 25	sylbi 119	1 |- ( A e. RR* -> -e -e A = A )

Syntax hints:   -> wi 4   \/ w3o 919   = wceq 1285   e. wcel 1434   RRcr 7252   +oocpnf 7422   -oocmnf 7423   RR*cxr 7424   -ucneg 7557   -ecxne 9135

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-pnf 7427  df-mnf 7428  df-xr 7429  df-sub 7558  df-neg 7559  df-xneg 9138

This theorem is referenced by:  xneg11  9191  xltneg  9193