Theorem xnegneg 10185

Description: Extended real version of negneg 8539. (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 10128	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 10182	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 10179	. . . . 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 8550	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 10182	. . . . 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 8276	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8591	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2271	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 10181	. . . 4 |- -e -oo = +oo
12		xnegeq 10179	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 10180	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2283	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 10179	. . . . 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 2293	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 10179	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2283	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 10179	. . . . 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 2293	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1340	. 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 1004   = wceq 1398   e. wcel 2205   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   -ucneg 8461   -ecxne 10121

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-sub 8462  df-neg 8463  df-xneg 10124

This theorem is referenced by:  xneg11  10186  xltneg  10188  xnegdi  10220  xnpcan  10224  xrnegiso  11972  infxrnegsupex  11973  xrnegcon1d  11974  xrminmax  11975  xrmin1inf  11977  xrmin2inf  11978  xrltmininf  11980  xrlemininf  11981  xrminltinf  11982  xrminadd  11985