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Theorem xnegneg 9609
Description: Extended real version of negneg 8005. (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 9556 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 9606 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 xnegeq 9603 . . . . 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 8016 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 rexneg 9606 . . . . 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 7746 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
98negnegd 8057 . . . 4  |-  ( A  e.  RR  ->  -u -u A  =  A )
104, 7, 93eqtrd 2174 . . 3  |-  ( A  e.  RR  ->  -e  -e A  =  A )
11 xnegmnf 9605 . . . 4  |-  -e -oo  = +oo
12 xnegeq 9603 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
13 xnegpnf 9604 . . . . . 6  |-  -e +oo  = -oo
1412, 13syl6eq 2186 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
15 xnegeq 9603 . . . . 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 2196 . . 3  |-  ( A  = +oo  ->  -e  -e A  =  A )
19 xnegeq 9603 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
2019, 11syl6eq 2186 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
21 xnegeq 9603 . . . . 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 2196 . . 3  |-  ( A  = -oo  ->  -e  -e A  =  A )
2510, 18, 243jaoi 1281 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -e  -e A  =  A )
261, 25sylbi 120 1  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 961    = wceq 1331    e. wcel 1480   RRcr 7612   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   -ucneg 7927    -ecxne 9549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-sub 7928  df-neg 7929  df-xneg 9552
This theorem is referenced by:  xneg11  9610  xltneg  9612  xnegdi  9644  xnpcan  9648  xrnegiso  11024  infxrnegsupex  11025  xrnegcon1d  11026  xrminmax  11027  xrmin1inf  11029  xrmin2inf  11030  xrltmininf  11032  xrlemininf  11033  xrminltinf  11034  xrminadd  11037
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