Theorem xnegneg 9802

Description: Extended real version of negneg 8181. (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 9745	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9799	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9796	. . . . 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 8192	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9799	. . . . 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 7919	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8233	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2212	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9798	. . . 4 |- -e -oo = +oo
12		xnegeq 9796	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9797	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2224	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9796	. . . . 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 2234	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9796	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2224	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9796	. . . . 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 2234	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1303	. 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 977   = wceq 1353   e. wcel 2146   RRcr 7785   +oocpnf 7963   -oocmnf 7964   RR*cxr 7965   -ucneg 8103   -ecxne 9738

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-sub 8104  df-neg 8105  df-xneg 9741

This theorem is referenced by:  xneg11  9803  xltneg  9805  xnegdi  9837  xnpcan  9841  xrnegiso  11236  infxrnegsupex  11237  xrnegcon1d  11238  xrminmax  11239  xrmin1inf  11241  xrmin2inf  11242  xrltmininf  11244  xrlemininf  11245  xrminltinf  11246  xrminadd  11249