Theorem xnegneg 9646

Description: Extended real version of negneg 8036. (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 9593	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9643	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9640	. . . . 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 8047	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9643	. . . . 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 7777	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8088	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2177	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9642	. . . 4 |- -e -oo = +oo
12		xnegeq 9640	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9641	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2189	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9640	. . . . 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 2199	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9640	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2189	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9640	. . . . 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 2199	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1282	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> -e -e A = A )
26	1, 25	sylbi 120	1 |- ( A e. RR* -> -e -e A = A )

Syntax hints:   -> wi 4   \/ w3o 962   = wceq 1332   e. wcel 1481   RRcr 7643   +oocpnf 7821   -oocmnf 7822   RR*cxr 7823   -ucneg 7958   -ecxne 9586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-sub 7959  df-neg 7960  df-xneg 9589

This theorem is referenced by:  xneg11  9647  xltneg  9649  xnegdi  9681  xnpcan  9685  xrnegiso  11063  infxrnegsupex  11064  xrnegcon1d  11065  xrminmax  11066  xrmin1inf  11068  xrmin2inf  11069  xrltmininf  11071  xrlemininf  11072  xrminltinf  11073  xrminadd  11076