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Theorem renegcli 9871
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 9873 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 9552 . 2  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 9571 . . . . 5  |-  ( x  e.  RR  ->  x  e.  CC )
4 df-neg 9799 . . . . . . 7  |-  -u A  =  ( 0  -  A )
54eqeq1i 2461 . . . . . 6  |-  ( -u A  =  x  <->  ( 0  -  A )  =  x )
6 0cn 9577 . . . . . . 7  |-  0  e.  CC
71recni 9597 . . . . . . 7  |-  A  e.  CC
8 subadd 9814 . . . . . . 7  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( 0  -  A
)  =  x  <->  ( A  +  x )  =  0 ) )
96, 7, 8mp3an12 1312 . . . . . 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 2537 . . . 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 2940 . 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 1398    e. wcel 1823   E.wrex 2805  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    - cmin 9796   -ucneg 9797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799
This theorem is referenced by:  resubcli  9872  renegcl  9873  recgt0ii  10446  inelr  10521  cju  10527  neg1rr  10636  sincos2sgn  14011  dvdslelem  14114  divalglem1  14136  divalglem6  14140  modsubi  14642  neghalfpire  23024  coseq0negpitopi  23062  pige3  23076  negpitopissre  23093  eff1o  23102  ellogrn  23113  logimclad  23126  logneg  23141  logcj  23159  argregt0  23163  argrege0  23164  argimgt0  23165  argimlt0  23166  logimul  23167  logneg2  23168  logcnlem3  23193  dvloglem  23197  logf1o2  23199  efopnlem2  23206  cxpsqrtlem  23251  abscxpbnd  23295  logreclem  23301  ang180lem2  23341  asinneg  23414  asinsin  23420  asin1  23422  asinrecl  23430  atanlogaddlem  23441  atanlogsublem  23443  atanlogsub  23444  atantan  23451  atanbndlem  23453  birthday  23482  ppiub  23677  lgsdir2lem1  23796  ex-fl  25370  normlem2  26226  logi  29362  cos2h  30286  tan2h  30287  fourierdlem5  32133  fourierdlem9  32137  fourierdlem18  32146  fourierdlem24  32152  fourierdlem38  32166  fourierdlem40  32168  fourierdlem43  32171  fourierdlem44  32172  fourierdlem46  32174  fourierdlem50  32178  fourierdlem62  32190  fourierdlem66  32194  fourierdlem74  32202  fourierdlem75  32203  fourierdlem76  32204  fourierdlem77  32205  fourierdlem78  32206  fourierdlem83  32211  fourierdlem85  32213  fourierdlem87  32215  fourierdlem88  32216  fourierdlem93  32221  fourierdlem94  32222  fourierdlem95  32223  fourierdlem101  32229  fourierdlem102  32230  fourierdlem103  32231  fourierdlem104  32232  fourierdlem111  32239  fourierdlem112  32240  fourierdlem113  32241  fourierdlem114  32242  sqwvfoura  32250  sqwvfourb  32251  fouriersw  32253  fouriercn  32254  bj-pinftyccb  35024  bj-minftyccb  35028  bj-pinftynminfty  35030  renegclALT  35091
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