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Theorem renegcl 7436
Description: Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
Assertion
Ref Expression
renegcl  |-  ( A  e.  RR  ->  -u A  e.  RR )

Proof of Theorem renegcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-rnegex 7147 . 2  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
2 recn 7168 . . . . 5  |-  ( x  e.  RR  ->  x  e.  CC )
3 df-neg 7349 . . . . . . 7  |-  -u A  =  ( 0  -  A )
43eqeq1i 2089 . . . . . 6  |-  ( -u A  =  x  <->  ( 0  -  A )  =  x )
5 recn 7168 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
6 0cn 7173 . . . . . . . 8  |-  0  e.  CC
7 subadd 7378 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
86, 7mp3an1 1256 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( 0  -  A )  =  x  <-> 
( A  +  x
)  =  0 ) )
95, 8sylan 277 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  CC )  ->  ( ( 0  -  A )  =  x  <-> 
( A  +  x
)  =  0 ) )
104, 9syl5bb 190 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  CC )  ->  ( -u A  =  x  <->  ( A  +  x )  =  0 ) )
112, 10sylan2 280 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u A  =  x  <->  ( A  +  x )  =  0 ) )
12 eleq1a 2151 . . . . 5  |-  ( x  e.  RR  ->  ( -u A  =  x  ->  -u A  e.  RR ) )
1312adantl 271 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( -u A  =  x  ->  -u A  e.  RR ) )
1411, 13sylbird 168 . . 3  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A  +  x )  =  0  ->  -u A  e.  RR ) )
1514rexlimdva 2478 . 2  |-  ( A  e.  RR  ->  ( E. x  e.  RR  ( A  +  x
)  =  0  ->  -u A  e.  RR ) )
161, 15mpd 13 1  |-  ( A  e.  RR  ->  -u A  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043    + caddc 7046    - cmin 7346   -ucneg 7347
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-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-setind 4288  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-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-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-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-sub 7348  df-neg 7349
This theorem is referenced by:  renegcli  7437  resubcl  7439  negreb  7440  renegcld  7551  negf1o  7553  ltnegcon1  7634  ltnegcon2  7635  lenegcon1  7637  lenegcon2  7638  mullt0  7651  recexre  7745  elnnz  8442  btwnz  8547  supinfneg  8764  infsupneg  8765  supminfex  8766  ublbneg  8779  negm  8781  rpnegap  8847  xnegcl  8975  xnegneg  8976  xltnegi  8978  iooneg  9086  iccneg  9087  icoshftf1o  9089  crim  9883  absnid  10097  absdiflt  10116  absdifle  10117  dfabsmax  10241  max0addsup  10243  negfi  10248  minmax  10250  min1inf  10251  min2inf  10252  infssuzex  10489
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