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Theorem renegcli 9666
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 9668 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypothesis
Ref Expression
renegcl.1  |-  A  e.  RR
Assertion
Ref Expression
renegcli  |-  -u A  e.  RR

Proof of Theorem renegcli
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 renegcl.1 . 2  |-  A  e.  RR
2 ax-rnegex 9349 . 2  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 9368 . . . . 5  |-  ( x  e.  RR  ->  x  e.  CC )
4 df-neg 9594 . . . . . . 7  |-  -u A  =  ( 0  -  A )
54eqeq1i 2448 . . . . . 6  |-  ( -u A  =  x  <->  ( 0  -  A )  =  x )
6 0cn 9374 . . . . . . 7  |-  0  e.  CC
71recni 9394 . . . . . . 7  |-  A  e.  CC
8 subadd 9609 . . . . . . 7  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
96, 7, 8mp3an12 1299 . . . . . 6  |-  ( x  e.  CC  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
105, 9syl5bb 257 . . . . 5  |-  ( x  e.  CC  ->  ( -u A  =  x  <->  ( A  +  x )  =  0 ) )
113, 10syl 16 . . . 4  |-  ( x  e.  RR  ->  ( -u A  =  x  <->  ( A  +  x )  =  0 ) )
12 eleq1a 2510 . . . 4  |-  ( x  e.  RR  ->  ( -u A  =  x  ->  -u A  e.  RR ) )
1311, 12sylbird 235 . . 3  |-  ( x  e.  RR  ->  (
( A  +  x
)  =  0  ->  -u A  e.  RR ) )
1413rexlimiv 2833 . 2  |-  ( E. x  e.  RR  ( A  +  x )  =  0  ->  -u A  e.  RR )
151, 2, 14mp2b 10 1  |-  -u A  e.  RR
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1364    e. wcel 1761   E.wrex 2714  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278    + caddc 9281    - cmin 9591   -ucneg 9592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-neg 9594
This theorem is referenced by:  resubcli  9667  renegcl  9668  recgt0ii  10234  inelr  10308  cju  10314  neg1rr  10422  sincos2sgn  13474  dvdslelem  13573  divalglem1  13594  divalglem6  13598  modsubi  14097  psgnodpmr  17979  neghalfpire  21886  coseq0negpitopi  21924  pige3  21938  negpitopissre  21955  eff1o  21964  ellogrn  21970  logimclad  21983  logneg  21995  logcj  22014  argregt0  22018  argrege0  22019  argimgt0  22020  argimlt0  22021  logimul  22022  logneg2  22023  logcnlem3  22048  dvloglem  22052  logf1o2  22054  efopnlem2  22061  cxpsqrlem  22106  abscxpbnd  22150  ang180lem2  22165  logreclem  22173  asinneg  22240  asinsin  22246  asin1  22248  asinrecl  22256  atanlogaddlem  22267  atanlogsublem  22269  atanlogsub  22270  atantan  22277  atanbndlem  22279  birthday  22307  ppiub  22502  lgsdir2lem1  22621  ex-fl  23589  normlem2  24448  cos2h  28348  tan2h  28349  bj-pinftyccb  32273  bj-minftyccb  32277  bj-pinftynminfty  32279  renegclALT  32336
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