Theorem xnegneg 10129

Description: Extended real version of negneg 8488. (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 10072	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 10126	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 10123	. . . . 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 8499	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 10126	. . . . 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 8225	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8540	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2268	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 10125	. . . 4 |- -e -oo = +oo
12		xnegeq 10123	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 10124	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2280	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 10123	. . . . 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 2290	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 10123	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2280	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 10123	. . . . 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 2290	. . 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 2202   RRcr 8091   +oocpnf 8270   -oocmnf 8271   RR*cxr 8272   -ucneg 8410   -ecxne 10065

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-sub 8411  df-neg 8412  df-xneg 10068

This theorem is referenced by:  xneg11  10130  xltneg  10132  xnegdi  10164  xnpcan  10168  xrnegiso  11902  infxrnegsupex  11903  xrnegcon1d  11904  xrminmax  11905  xrmin1inf  11907  xrmin2inf  11908  xrltmininf  11910  xrlemininf  11911  xrminltinf  11912  xrminadd  11915