Theorem xnegneg 10068

Description: Extended real version of negneg 8429. (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 10011	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 10065	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 10062	. . . . 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 8440	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 10065	. . . . 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 8165	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 8481	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2268	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 10064	. . . 4 |- -e -oo = +oo
12		xnegeq 10062	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 10063	. . . . . 6 |- -e +oo = -oo
14	12, 13	eqtrdi 2280	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 10062	. . . . 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 10062	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	eqtrdi 2280	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 10062	. . . . 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 1339	. 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 1003   = wceq 1397   e. wcel 2202   RRcr 8031   +oocpnf 8211   -oocmnf 8212   RR*cxr 8213   -ucneg 8351   -ecxne 10004

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-sub 8352  df-neg 8353  df-xneg 10007

This theorem is referenced by:  xneg11  10069  xltneg  10071  xnegdi  10103  xnpcan  10107  xrnegiso  11840  infxrnegsupex  11841  xrnegcon1d  11842  xrminmax  11843  xrmin1inf  11845  xrmin2inf  11846  xrltmininf  11848  xrlemininf  11849  xrminltinf  11850  xrminadd  11853