Theorem xnegneg 10214

Description: Extended real version of negneg 8566. (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 10157	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 10211	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 10208	. . . . 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 8577	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 10211	. . . . 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 8302	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8618	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2275	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 10210	. . . 4 |- -e -oo = +oo
12		xnegeq 10208	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 10209	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2287	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 10208	. . . . 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 2297	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 10208	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2287	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 10208	. . . . 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 2297	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1344	. 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 1008   = wceq 1402   e. wcel 2209   RRcr 8168   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349   -ucneg 8488   -ecxne 10150

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-sub 8489  df-neg 8490  df-xneg 10153

This theorem is referenced by:  xneg11  10215  xltneg  10217  xnegdi  10249  xnpcan  10253  xrnegiso  12006  infxrnegsupex  12007  xrnegcon1d  12008  xrminmax  12009  xrmin1inf  12011  xrmin2inf  12012  xrltmininf  12014  xrlemininf  12015  xrminltinf  12016  xrminadd  12019