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Theorem xnegneg 9190
Description: Extended real version of negneg 7635. (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 9142 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 9187 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 xnegeq 9184 . . . . 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 7646 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
6 rexneg 9187 . . . . 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 7378 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
98negnegd 7687 . . . 4  |-  ( A  e.  RR  ->  -u -u A  =  A )
104, 7, 93eqtrd 2119 . . 3  |-  ( A  e.  RR  ->  -e  -e A  =  A )
11 xnegmnf 9186 . . . 4  |-  -e -oo  = +oo
12 xnegeq 9184 . . . . . 6  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
13 xnegpnf 9185 . . . . . 6  |-  -e +oo  = -oo
1412, 13syl6eq 2131 . . . . 5  |-  ( A  = +oo  ->  -e
A  = -oo )
15 xnegeq 9184 . . . . 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 2141 . . 3  |-  ( A  = +oo  ->  -e  -e A  =  A )
19 xnegeq 9184 . . . . . 6  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
2019, 11syl6eq 2131 . . . . 5  |-  ( A  = -oo  ->  -e
A  = +oo )
21 xnegeq 9184 . . . . 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 2141 . . 3  |-  ( A  = -oo  ->  -e  -e A  =  A )
2510, 18, 243jaoi 1235 . 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 919    = wceq 1285    e. wcel 1434   RRcr 7252   +oocpnf 7422   -oocmnf 7423   RR*cxr 7424   -ucneg 7557    -ecxne 9135
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-pnf 7427  df-mnf 7428  df-xr 7429  df-sub 7558  df-neg 7559  df-xneg 9138
This theorem is referenced by:  xneg11  9191  xltneg  9193
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