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Theorem renegcli 9879
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 9881 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 9562 . 2  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 9581 . . . . 5  |-  ( x  e.  RR  ->  x  e.  CC )
4 df-neg 9807 . . . . . . 7  |-  -u A  =  ( 0  -  A )
54eqeq1i 2474 . . . . . 6  |-  ( -u A  =  x  <->  ( 0  -  A )  =  x )
6 0cn 9587 . . . . . . 7  |-  0  e.  CC
71recni 9607 . . . . . . 7  |-  A  e.  CC
8 subadd 9822 . . . . . . 7  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
96, 7, 8mp3an12 1314 . . . . . 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 2550 . . . 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 2949 . 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 1379    e. wcel 1767   E.wrex 2815  (class class class)co 6283   CCcc 9489   RRcr 9490   0cc0 9491    + caddc 9494    - cmin 9804   -ucneg 9805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-ltxr 9632  df-sub 9806  df-neg 9807
This theorem is referenced by:  resubcli  9880  renegcl  9881  recgt0ii  10450  inelr  10525  cju  10531  neg1rr  10639  sincos2sgn  13789  dvdslelem  13888  divalglem1  13910  divalglem6  13914  modsubi  14416  psgnodpmr  18409  neghalfpire  22607  coseq0negpitopi  22645  pige3  22659  negpitopissre  22676  eff1o  22685  ellogrn  22691  logimclad  22704  logneg  22716  logcj  22735  argregt0  22739  argrege0  22740  argimgt0  22741  argimlt0  22742  logimul  22743  logneg2  22744  logcnlem3  22769  dvloglem  22773  logf1o2  22775  efopnlem2  22782  cxpsqrtlem  22827  abscxpbnd  22871  ang180lem2  22886  logreclem  22894  asinneg  22961  asinsin  22967  asin1  22969  asinrecl  22977  atanlogaddlem  22988  atanlogsublem  22990  atanlogsub  22991  atantan  22998  atanbndlem  23000  birthday  23028  ppiub  23223  lgsdir2lem1  23342  ex-fl  24861  normlem2  25720  cos2h  29639  tan2h  29640  fourierdlem5  31428  fourierdlem9  31432  fourierdlem18  31441  fourierdlem24  31447  fourierdlem38  31461  fourierdlem40  31463  fourierdlem43  31466  fourierdlem44  31467  fourierdlem46  31469  fourierdlem50  31473  fourierdlem62  31485  fourierdlem66  31489  fourierdlem74  31497  fourierdlem75  31498  fourierdlem76  31499  fourierdlem77  31500  fourierdlem78  31501  fourierdlem83  31506  fourierdlem85  31508  fourierdlem87  31510  fourierdlem88  31511  fourierdlem93  31516  fourierdlem94  31517  fourierdlem95  31518  fourierdlem101  31524  fourierdlem102  31525  fourierdlem103  31526  fourierdlem104  31527  fourierdlem111  31534  fourierdlem112  31535  fourierdlem113  31536  fourierdlem114  31537  sqwvfoura  31545  sqwvfourb  31546  fouriersw  31548  fouriercn  31549  bj-pinftyccb  33705  bj-minftyccb  33709  bj-pinftynminfty  33711  renegclALT  33775
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