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Theorem xnegcl 8745
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 8696 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 rexneg 8743 . . . . 5  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
3 renegcl 7272 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
42, 3eqeltrd 2114 . . . 4  |-  ( A  e.  RR  ->  -e
A  e.  RR )
54rexrd 7075 . . 3  |-  ( A  e.  RR  ->  -e
A  e.  RR* )
6 xnegeq 8740 . . . 4  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
7 xnegpnf 8741 . . . . 5  |-  -e +oo  = -oo
8 mnfxr 8694 . . . . 5  |- -oo  e.  RR*
97, 8eqeltri 2110 . . . 4  |-  -e +oo  e.  RR*
106, 9syl6eqel 2128 . . 3  |-  ( A  = +oo  ->  -e
A  e.  RR* )
11 xnegeq 8740 . . . 4  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
12 xnegmnf 8742 . . . . 5  |-  -e -oo  = +oo
13 pnfxr 8692 . . . . 5  |- +oo  e.  RR*
1412, 13eqeltri 2110 . . . 4  |-  -e -oo  e.  RR*
1511, 14syl6eqel 2128 . . 3  |-  ( A  = -oo  ->  -e
A  e.  RR* )
165, 10, 153jaoi 1198 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -e
A  e.  RR* )
171, 16sylbi 114 1  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ w3o 884    = wceq 1243    e. wcel 1393   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059   -ucneg 7183    -ecxne 8686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-sub 7184  df-neg 7185  df-xneg 8689
This theorem is referenced by:  xltneg  8749  xleneg  8750  xnegcld  8755
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