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Theorem xnegneg 9865
Description: Extended real version of negneg 8238. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegneg  |-  ( A  e.  RR*  ->  -e  -e A  =  A )

Proof of Theorem xnegneg
StepHypRef Expression
1 elxr 9808 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 9862 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 xnegeq 9859 . . . . 5  |-  (  -e A  =  -u A  -> 
-e  -e
A  =  -e -u A )
42, 3syl 14 . . . 4  |-  ( A  e.  RR  ->  -e  -e A  =  -e -u A )
5 renegcl 8249 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 rexneg 9862 . . . . 5  |-  ( -u A  e.  RR  ->  -e -u A  =  -u -u A )
75, 6syl 14 . . . 4  |-  ( A  e.  RR  ->  -e -u A  =  -u -u A
)
8 recn 7975 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
98negnegd 8290 . . . 4  |-  ( A  e.  RR  ->  -u -u A  =  A )
104, 7, 93eqtrd 2226 . . 3  |-  ( A  e.  RR  ->  -e  -e A  =  A )
11 xnegmnf 9861 . . . 4  |-  -e -oo  = +oo
12 xnegeq 9859 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
13 xnegpnf 9860 . . . . . 6  |-  -e +oo  = -oo
1412, 13eqtrdi 2238 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
15 xnegeq 9859 . . . . 5  |-  (  -e A  = -oo  -> 
-e  -e
A  =  -e -oo )
1614, 15syl 14 . . . 4  |-  ( A  = +oo  ->  -e  -e A  =  -e -oo )
17 id 19 . . . 4  |-  ( A  = +oo  ->  A  = +oo )
1811, 16, 173eqtr4a 2248 . . 3  |-  ( A  = +oo  ->  -e  -e A  =  A )
19 xnegeq 9859 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
2019, 11eqtrdi 2238 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
21 xnegeq 9859 . . . . 5  |-  (  -e A  = +oo  -> 
-e  -e
A  =  -e +oo )
2220, 21syl 14 . . . 4  |-  ( A  = -oo  ->  -e  -e A  =  -e +oo )
23 id 19 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
2413, 22, 233eqtr4a 2248 . . 3  |-  ( A  = -oo  ->  -e  -e A  =  A )
2510, 18, 243jaoi 1314 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -e  -e A  =  A )
261, 25sylbi 121 1  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 979    = wceq 1364    e. wcel 2160   RRcr 7841   +oocpnf 8020   -oocmnf 8021   RR*cxr 8022   -ucneg 8160    -ecxne 9801
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-sub 8161  df-neg 8162  df-xneg 9804
This theorem is referenced by:  xneg11  9866  xltneg  9868  xnegdi  9900  xnpcan  9904  xrnegiso  11305  infxrnegsupex  11306  xrnegcon1d  11307  xrminmax  11308  xrmin1inf  11310  xrmin2inf  11311  xrltmininf  11313  xrlemininf  11314  xrminltinf  11315  xrminadd  11318
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