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Theorem cnegex 9009
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 8850 . 2  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 ax-rnegex 8824 . . . . . . 7  |-  ( a  e.  RR  ->  E. c  e.  RR  ( a  +  c )  =  0 )
3 ax-rnegex 8824 . . . . . . 7  |-  ( b  e.  RR  ->  E. d  e.  RR  ( b  +  d )  =  0 )
42, 3anim12i 549 . . . . . 6  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( E. c  e.  RR  ( a  +  c )  =  0  /\  E. d  e.  RR  ( b  +  d )  =  0 ) )
5 reeanv 2720 . . . . . 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 203 . . . . 5  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  E. c  e.  RR  E. d  e.  RR  (
( a  +  c )  =  0  /\  ( b  +  d )  =  0 ) )
7 ax-icn 8812 . . . . . . . . . . 11  |-  _i  e.  CC
87a1i 10 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  _i  e.  CC )
9 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  RR )
109recnd 8877 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  CC )
118, 10mulcld 8871 . . . . . . . . 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 736 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  RR )
1312recnd 8877 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  CC )
1411, 13addcld 8870 . . . . . . . 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 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  RR )
1615recnd 8877 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  CC )
17 simpllr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  RR )
1817recnd 8877 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  CC )
198, 18mulcld 8871 . . . . . . . . . . . . 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 8873 . . . . . . . . . . . 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 8875 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( b  +  d )  =  0 )
2322oveq2d 5890 . . . . . . . . . . . . . . 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 9007 . . . . . . . . . . . . . . . 16  |-  ( _i  e.  CC  ->  (
_i  x.  0 )  =  0 )
257, 24ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( _i  x.  0 )  =  0
2623, 25syl6eq 2344 . . . . . . . . . . . . . 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 2330 . . . . . . . . . . . . 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 5890 . . . . . . . . . . . 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 9008 . . . . . . . . . . . . 13  |-  ( a  e.  CC  ->  (
a  +  0 )  =  a )
3016, 29syl 15 . . . . . . . . . . . 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 2332 . . . . . . . . . . 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 5889 . . . . . . . . . 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 8870 . . . . . . . . . . 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 8873 . . . . . . . . . 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 2330 . . . . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  c )  =  0 )
3735, 36eqtr3d 2330 . . . . . . . 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 5882 . . . . . . . . . 10  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( a  +  ( _i  x.  b ) )  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
3938eqeq1d 2304 . . . . . . . . 9  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( ( a  +  ( _i  x.  b
) )  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) )  =  0 ) )
4039rspcev 2897 . . . . . . . 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 642 . . . . . . 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 423 . . . . . 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 2687 . . . . 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 14 . . . 4  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  +  x
)  =  0 )
45 oveq1 5881 . . . . . 6  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  x ) )
4645eqeq1d 2304 . . . . 5  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4746rexbidv 2577 . . . 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 213 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( A  =  ( a  +  ( _i  x.  b ) )  ->  E. x  e.  CC  ( A  +  x
)  =  0 ) )
4948rexlimivv 2685 . 2  |-  ( E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b
) )  ->  E. x  e.  CC  ( A  +  x )  =  0 )
501, 49syl 15 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   _ici 8755    + caddc 8756    x. cmul 8758
This theorem is referenced by:  addid2  9011  addcan2  9013  0cnALT  9057  negeu  9058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888
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