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Theorem renegcli 9682
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 9684 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 9365 . 2  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 9384 . . . . 5  |-  ( x  e.  RR  ->  x  e.  CC )
4 df-neg 9610 . . . . . . 7  |-  -u A  =  ( 0  -  A )
54eqeq1i 2450 . . . . . 6  |-  ( -u A  =  x  <->  ( 0  -  A )  =  x )
6 0cn 9390 . . . . . . 7  |-  0  e.  CC
71recni 9410 . . . . . . 7  |-  A  e.  CC
8 subadd 9625 . . . . . . 7  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
96, 7, 8mp3an12 1304 . . . . . 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 2512 . . . 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 2847 . 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 1369    e. wcel 1756   E.wrex 2728  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294    + caddc 9297    - cmin 9607   -ucneg 9608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-ltxr 9435  df-sub 9609  df-neg 9610
This theorem is referenced by:  resubcli  9683  renegcl  9684  recgt0ii  10250  inelr  10324  cju  10330  neg1rr  10438  sincos2sgn  13490  dvdslelem  13589  divalglem1  13610  divalglem6  13614  modsubi  14113  psgnodpmr  18032  neghalfpire  21939  coseq0negpitopi  21977  pige3  21991  negpitopissre  22008  eff1o  22017  ellogrn  22023  logimclad  22036  logneg  22048  logcj  22067  argregt0  22071  argrege0  22072  argimgt0  22073  argimlt0  22074  logimul  22075  logneg2  22076  logcnlem3  22101  dvloglem  22105  logf1o2  22107  efopnlem2  22114  cxpsqrlem  22159  abscxpbnd  22203  ang180lem2  22218  logreclem  22226  asinneg  22293  asinsin  22299  asin1  22301  asinrecl  22309  atanlogaddlem  22320  atanlogsublem  22322  atanlogsub  22323  atantan  22330  atanbndlem  22332  birthday  22360  ppiub  22555  lgsdir2lem1  22674  ex-fl  23666  normlem2  24525  cos2h  28435  tan2h  28436  bj-pinftyccb  32556  bj-minftyccb  32560  bj-pinftynminfty  32562  renegclALT  32626
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