Theorem xnegneg 9769

Description: Extended real version of negneg 8148. (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 9712	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9766	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9763	. . . . 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 8159	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9766	. . . . 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 7886	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8200	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2202	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9765	. . . 4 |- -e -oo = +oo
12		xnegeq 9763	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9764	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2215	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9763	. . . . 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 2225	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9763	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2215	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9763	. . . . 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 2225	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1293	. 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 967   = wceq 1343   e. wcel 2136   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   -ucneg 8070   -ecxne 9705

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-sub 8071  df-neg 8072  df-xneg 9708

This theorem is referenced by:  xneg11  9770  xltneg  9772  xnegdi  9804  xnpcan  9808  xrnegiso  11203  infxrnegsupex  11204  xrnegcon1d  11205  xrminmax  11206  xrmin1inf  11208  xrmin2inf  11209  xrltmininf  11211  xrlemininf  11212  xrminltinf  11213  xrminadd  11216