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Theorem xnegneg 9293
Description: Extended real version of negneg 7730. (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 9245 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 9290 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 xnegeq 9287 . . . . 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 7741 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 rexneg 9290 . . . . 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 7473 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
98negnegd 7782 . . . 4  |-  ( A  e.  RR  ->  -u -u A  =  A )
104, 7, 93eqtrd 2124 . . 3  |-  ( A  e.  RR  ->  -e  -e A  =  A )
11 xnegmnf 9289 . . . 4  |-  -e -oo  = +oo
12 xnegeq 9287 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
13 xnegpnf 9288 . . . . . 6  |-  -e +oo  = -oo
1412, 13syl6eq 2136 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
15 xnegeq 9287 . . . . 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 2146 . . 3  |-  ( A  = +oo  ->  -e  -e A  =  A )
19 xnegeq 9287 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
2019, 11syl6eq 2136 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
21 xnegeq 9287 . . . . 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 2146 . . 3  |-  ( A  = -oo  ->  -e  -e A  =  A )
2510, 18, 243jaoi 1239 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -e  -e A  =  A )
261, 25sylbi 119 1  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 923    = wceq 1289    e. wcel 1438   RRcr 7347   +oocpnf 7517   -oocmnf 7518   RR*cxr 7519   -ucneg 7652    -ecxne 9238
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-sub 7653  df-neg 7654  df-xneg 9241
This theorem is referenced by:  xneg11  9294  xltneg  9296
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