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Theorem cnegex 9764
Description: Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
cnegex  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
Distinct variable group:    x, A

Proof of Theorem cnegex
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 9595 . 2  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 ax-rnegex 9566 . . . . . . 7  |-  ( a  e.  RR  ->  E. c  e.  RR  ( a  +  c )  =  0 )
3 ax-rnegex 9566 . . . . . . 7  |-  ( b  e.  RR  ->  E. d  e.  RR  ( b  +  d )  =  0 )
42, 3anim12i 566 . . . . . 6  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( E. c  e.  RR  ( a  +  c )  =  0  /\  E. d  e.  RR  ( b  +  d )  =  0 ) )
5 reeanv 3011 . . . . . 6  |-  ( E. c  e.  RR  E. d  e.  RR  (
( a  +  c )  =  0  /\  ( b  +  d )  =  0 )  <-> 
( E. c  e.  RR  ( a  +  c )  =  0  /\  E. d  e.  RR  ( b  +  d )  =  0 ) )
64, 5sylibr 212 . . . . 5  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  E. c  e.  RR  E. d  e.  RR  (
( a  +  c )  =  0  /\  ( b  +  d )  =  0 ) )
7 ax-icn 9554 . . . . . . . . . . 11  |-  _i  e.  CC
87a1i 11 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  _i  e.  CC )
9 simplrr 762 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  RR )
109recnd 9625 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  CC )
118, 10mulcld 9619 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( _i  x.  d )  e.  CC )
12 simplrl 761 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  RR )
1312recnd 9625 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  CC )
1411, 13addcld 9618 . . . . . . . 8  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( _i  x.  d )  +  c )  e.  CC )
15 simplll 759 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  RR )
1615recnd 9625 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  CC )
17 simpllr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  RR )
1817recnd 9625 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  CC )
198, 18mulcld 9619 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( _i  x.  b )  e.  CC )
2016, 19, 11addassd 9621 . . . . . . . . . . . 12  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( a  +  ( _i  x.  b ) )  +  ( _i  x.  d
) )  =  ( a  +  ( ( _i  x.  b )  +  ( _i  x.  d ) ) ) )
218, 18, 10adddid 9623 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( _i  x.  ( b  +  d ) )  =  ( ( _i  x.  b
)  +  ( _i  x.  d ) ) )
22 simprr 757 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( b  +  d )  =  0 )
2322oveq2d 6297 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( _i  x.  ( b  +  d ) )  =  ( _i  x.  0 ) )
24 mul01 9762 . . . . . . . . . . . . . . . 16  |-  ( _i  e.  CC  ->  (
_i  x.  0 )  =  0 )
257, 24ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( _i  x.  0 )  =  0
2623, 25syl6eq 2500 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( _i  x.  ( b  +  d ) )  =  0 )
2721, 26eqtr3d 2486 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( _i  x.  b )  +  ( _i  x.  d
) )  =  0 )
2827oveq2d 6297 . . . . . . . . . . . 12  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  ( ( _i  x.  b )  +  ( _i  x.  d ) ) )  =  ( a  +  0 ) )
29 addid1 9763 . . . . . . . . . . . . 13  |-  ( a  e.  CC  ->  (
a  +  0 )  =  a )
3016, 29syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  0 )  =  a )
3120, 28, 303eqtrd 2488 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( a  +  ( _i  x.  b ) )  +  ( _i  x.  d
) )  =  a )
3231oveq1d 6296 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( ( a  +  ( _i  x.  b ) )  +  ( _i  x.  d ) )  +  c )  =  ( a  +  c ) )
3316, 19addcld 9618 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  ( _i  x.  b
) )  e.  CC )
3433, 11, 13addassd 9621 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( ( a  +  ( _i  x.  b ) )  +  ( _i  x.  d ) )  +  c )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
3532, 34eqtr3d 2486 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  c )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
36 simprl 756 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  c )  =  0 )
3735, 36eqtr3d 2486 . . . . . . . 8  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) )  =  0 )
38 oveq2 6289 . . . . . . . . . 10  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( a  +  ( _i  x.  b ) )  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
3938eqeq1d 2445 . . . . . . . . 9  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( ( a  +  ( _i  x.  b
) )  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) )  =  0 ) )
4039rspcev 3196 . . . . . . . 8  |-  ( ( ( ( _i  x.  d )  +  c )  e.  CC  /\  ( ( a  +  ( _i  x.  b
) )  +  ( ( _i  x.  d
)  +  c ) )  =  0 )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  +  x
)  =  0 )
4114, 37, 40syl2anc 661 . . . . . . 7  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  +  x
)  =  0 )
4241ex 434 . . . . . 6  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( c  e.  RR  /\  d  e.  RR ) )  -> 
( ( ( a  +  c )  =  0  /\  ( b  +  d )  =  0 )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4342rexlimdvva 2942 . . . . 5  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( E. c  e.  RR  E. d  e.  RR  ( ( a  +  c )  =  0  /\  ( b  +  d )  =  0 )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
446, 43mpd 15 . . . 4  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  +  x
)  =  0 )
45 oveq1 6288 . . . . . 6  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  x ) )
4645eqeq1d 2445 . . . . 5  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4746rexbidv 2954 . . . 4  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( E. x  e.  CC  ( A  +  x
)  =  0  <->  E. x  e.  CC  (
( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4844, 47syl5ibrcom 222 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( A  =  ( a  +  ( _i  x.  b ) )  ->  E. x  e.  CC  ( A  +  x
)  =  0 ) )
4948rexlimivv 2940 . 2  |-  ( E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b
) )  ->  E. x  e.  CC  ( A  +  x )  =  0 )
501, 49syl 16 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   E.wrex 2794  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   _ici 9497    + caddc 9498    x. cmul 9500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636
This theorem is referenced by:  addid2  9766  addcan2  9768  0cnALT  9814  negeu  9815
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