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Theorem renegcl 7722
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 7433 . 2  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
2 recn 7454 . . . . 5  |-  ( x  e.  RR  ->  x  e.  CC )
3 df-neg 7635 . . . . . . 7  |-  -u A  =  ( 0  -  A )
43eqeq1i 2095 . . . . . 6  |-  ( -u A  =  x  <->  ( 0  -  A )  =  x )
5 recn 7454 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
6 0cn 7459 . . . . . . . 8  |-  0  e.  CC
7 subadd 7664 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
86, 7mp3an1 1260 . . . . . . 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 2159 . . . . 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 2489 . 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 1289    e. wcel 1438   E.wrex 2360  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329    + caddc 7332    - cmin 7632   -ucneg 7633
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  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-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-neg 7635
This theorem is referenced by:  renegcli  7723  resubcl  7725  negreb  7726  renegcld  7837  negf1o  7839  ltnegcon1  7920  ltnegcon2  7921  lenegcon1  7923  lenegcon2  7924  mullt0  7937  recexre  8031  elnnz  8730  btwnz  8835  supinfneg  9052  infsupneg  9053  supminfex  9054  ublbneg  9067  negm  9069  rpnegap  9135  xnegcl  9263  xnegneg  9264  xltnegi  9266  iooneg  9374  iccneg  9375  icoshftf1o  9377  crim  10257  absnid  10471  absdiflt  10490  absdifle  10491  dfabsmax  10615  max0addsup  10617  negfi  10623  minmax  10625  min1inf  10626  min2inf  10627  infssuzex  11038
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