Theorem xnegneg 9865

Description: Extended real version of negneg 8238. (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 9808	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9862	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9859	. . . . 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 8249	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9862	. . . . 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 7975	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8290	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2226	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9861	. . . 4 |- -e -oo = +oo
12		xnegeq 9859	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9860	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2238	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9859	. . . . 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 2248	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9859	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2238	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9859	. . . . 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 2248	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1314	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> -e -e A = A )
26	1, 25	sylbi 121	1 |- ( A e. RR* -> -e -e A = A )

Syntax hints:   -> wi 4   \/ w3o 979   = wceq 1364   e. wcel 2160   RRcr 7841   +oocpnf 8020   -oocmnf 8021   RR*cxr 8022   -ucneg 8160   -ecxne 9801

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-sub 8161  df-neg 8162  df-xneg 9804

This theorem is referenced by:  xneg11  9866  xltneg  9868  xnegdi  9900  xnpcan  9904  xrnegiso  11305  infxrnegsupex  11306  xrnegcon1d  11307  xrminmax  11308  xrmin1inf  11310  xrmin2inf  11311  xrltmininf  11313  xrlemininf  11314  xrminltinf  11315  xrminadd  11318