Theorem xnegneg 9910

Description: Extended real version of negneg 8278. (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 9853	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9907	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9904	. . . . 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 8289	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9907	. . . . 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 8014	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8330	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2233	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9906	. . . 4 |- -e -oo = +oo
12		xnegeq 9904	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9905	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2245	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9904	. . . . 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 2255	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9904	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2245	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9904	. . . . 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 2255	. . 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 2167   RRcr 7880   +oocpnf 8060   -oocmnf 8061   RR*cxr 8062   -ucneg 8200   -ecxne 9846

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-distr 7985  ax-i2m1 7986  ax-0id 7989  ax-rnegex 7990  ax-cnre 7992

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-pnf 8065  df-mnf 8066  df-xr 8067  df-sub 8201  df-neg 8202  df-xneg 9849

This theorem is referenced by:  xneg11  9911  xltneg  9913  xnegdi  9945  xnpcan  9949  xrnegiso  11429  infxrnegsupex  11430  xrnegcon1d  11431  xrminmax  11432  xrmin1inf  11434  xrmin2inf  11435  xrltmininf  11437  xrlemininf  11438  xrminltinf  11439  xrminadd  11442