Theorem xnegneg 9835

Description: Extended real version of negneg 8209. (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 9778	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 9832	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 9829	. . . . 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 8220	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 9832	. . . . 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 7946	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8261	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2214	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 9831	. . . 4 |- -e -oo = +oo
12		xnegeq 9829	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 9830	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2226	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 9829	. . . . 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 2236	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 9829	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2226	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 9829	. . . . 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 2236	. . 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 2148   RRcr 7812   +oocpnf 7991   -oocmnf 7992   RR*cxr 7993   -ucneg 8131   -ecxne 9771

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-sub 8132  df-neg 8133  df-xneg 9774

This theorem is referenced by:  xneg11  9836  xltneg  9838  xnegdi  9870  xnpcan  9874  xrnegiso  11272  infxrnegsupex  11273  xrnegcon1d  11274  xrminmax  11275  xrmin1inf  11277  xrmin2inf  11278  xrltmininf  11280  xrlemininf  11281  xrminltinf  11282  xrminadd  11285