Theorem xnegneg 8900

Description: Extended real version of negneg 7358. (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 8850	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		rexneg 8897	. . . . 5 |- ( A e. RR -> -e A = -u A )
3		xnegeq 8894	. . . . 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 7369	. . . . 5 |- ( A e. RR -> -u A e. RR )
6		rexneg 8897	. . . . 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 7106	. . . . 5 |- ( A e. RR -> A e. CC )
9	8	negnegd 7410	. . . 4 |- ( A e. RR -> -u -u A = A )
10	4, 7, 9	3eqtrd 2117	. . 3 |- ( A e. RR -> -e -e A = A )
11		xnegmnf 8896	. . . 4 |- -e -oo = +oo
12		xnegeq 8894	. . . . . 6 |- ( A = +oo -> -e A = -e +oo )
13		xnegpnf 8895	. . . . . 6 |- -e +oo = -oo
14	12, 13	syl6eq 2129	. . . . 5 |- ( A = +oo -> -e A = -oo )
15		xnegeq 8894	. . . . 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 2139	. . 3 |- ( A = +oo -> -e -e A = A )
19		xnegeq 8894	. . . . . 6 |- ( A = -oo -> -e A = -e -oo )
20	19, 11	syl6eq 2129	. . . . 5 |- ( A = -oo -> -e A = +oo )
21		xnegeq 8894	. . . . 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 2139	. . 3 |- ( A = -oo -> -e -e A = A )
25	10, 18, 24	3jaoi 1234	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> -e -e A = A )
26	1, 25	sylbi 119	1 |- ( A e. RR* -> -e -e A = A )

Syntax hints:   -> wi 4   \/ w3o 918   = wceq 1284   e. wcel 1433   RRcr 6980   +oocpnf 7150   -oocmnf 7151   RR*cxr 7152   -ucneg 7280   -ecxne 8840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-sub 7281  df-neg 7282  df-xneg 8843

This theorem is referenced by:  xneg11  8901  xltneg  8903