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Theorem xnegcl 8975
Description: Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegcl  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )

Proof of Theorem xnegcl
StepHypRef Expression
1 elxr 8928 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 8973 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 renegcl 7436 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
42, 3eqeltrd 2156 . . . 4  |-  ( A  e.  RR  ->  -e
A  e.  RR )
54rexrd 7230 . . 3  |-  ( A  e.  RR  ->  -e
A  e.  RR* )
6 xnegeq 8970 . . . 4  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
7 xnegpnf 8971 . . . . 5  |-  -e +oo  = -oo
8 mnfxr 7237 . . . . 5  |- -oo  e.  RR*
97, 8eqeltri 2152 . . . 4  |-  -e +oo  e.  RR*
106, 9syl6eqel 2170 . . 3  |-  ( A  = +oo  ->  -e
A  e.  RR* )
11 xnegeq 8970 . . . 4  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
12 xnegmnf 8972 . . . . 5  |-  -e -oo  = +oo
13 pnfxr 7233 . . . . 5  |- +oo  e.  RR*
1412, 13eqeltri 2152 . . . 4  |-  -e -oo  e.  RR*
1511, 14syl6eqel 2170 . . 3  |-  ( A  = -oo  ->  -e
A  e.  RR* )
165, 10, 153jaoi 1235 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -e
A  e.  RR* )
171, 16sylbi 119 1  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 919    = wceq 1285    e. wcel 1434   RRcr 7042   +oocpnf 7212   -oocmnf 7213   RR*cxr 7214   -ucneg 7347    -ecxne 8921
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-sub 7348  df-neg 7349  df-xneg 8924
This theorem is referenced by:  xltneg  8979  xleneg  8980  xnegcld  8985
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