Theorem xnegneg 9616

Description: Extended real version of negneg 8012. (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 9563	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9613	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9610	. . . . 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 8023	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9613	. . . . 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 7753	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8064	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2176	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9612	. . . 4 |- -e -oo = +oo
12		xnegeq 9610	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9611	. . . . . 6 |- -e +oo = -oo
14	12, 13	syl6eq 2188	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9610	. . . . 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 2198	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9610	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	syl6eq 2188	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9610	. . . . 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 2198	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1281	. 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 961   = wceq 1331   e. wcel 1480   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   -ucneg 7934   -ecxne 9556

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-sub 7935  df-neg 7936  df-xneg 9559

This theorem is referenced by:  xneg11  9617  xltneg  9619  xnegdi  9651  xnpcan  9655  xrnegiso  11031  infxrnegsupex  11032  xrnegcon1d  11033  xrminmax  11034  xrmin1inf  11036  xrmin2inf  11037  xrltmininf  11039  xrlemininf  11040  xrminltinf  11041  xrminadd  11044