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Theorem cnegex 8988
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
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.

Proof of Theorem cnegex
StepHypRef Expression
1 cnre 8829 . 2  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 ax-rnegex 8803 . . . . . . 7  |-  ( a  e.  RR  ->  E. c  e.  RR  ( a  +  c )  =  0 )
3 ax-rnegex 8803 . . . . . . 7  |-  ( b  e.  RR  ->  E. d  e.  RR  ( b  +  d )  =  0 )
42, 3anim12i 551 . . . . . 6  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( E. c  e.  RR  ( a  +  c )  =  0  /\  E. d  e.  RR  ( b  +  d )  =  0 ) )
5 reeanv 2708 . . . . . 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 205 . . . . 5  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  E. c  e.  RR  E. d  e.  RR  (
( a  +  c )  =  0  /\  ( b  +  d )  =  0 ) )
7 ax-icn 8791 . . . . . . . . . . 11  |-  _i  e.  CC
87a1i 12 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  _i  e.  CC )
9 simplrr 739 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  RR )
109recnd 8856 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  CC )
118, 10mulcld 8850 . . . . . . . . 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 738 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  RR )
1312recnd 8856 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  CC )
1411, 13addcld 8849 . . . . . . . 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 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  RR )
1615recnd 8856 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  CC )
17 simpllr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  RR )
1817recnd 8856 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  CC )
198, 18mulcld 8850 . . . . . . . . . . . . 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 8852 . . . . . . . . . . . 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 8854 . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( b  +  d )  =  0 )
2322oveq2d 5835 . . . . . . . . . . . . . . 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 8986 . . . . . . . . . . . . . . . 16  |-  ( _i  e.  CC  ->  (
_i  x.  0 )  =  0 )
257, 24ax-mp 10 . . . . . . . . . . . . . . 15  |-  ( _i  x.  0 )  =  0
2623, 25syl6eq 2332 . . . . . . . . . . . . . 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 2318 . . . . . . . . . . . . 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 5835 . . . . . . . . . . . 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 8987 . . . . . . . . . . . . 13  |-  ( a  e.  CC  ->  (
a  +  0 )  =  a )
3016, 29syl 17 . . . . . . . . . . . 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 2320 . . . . . . . . . . 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 5834 . . . . . . . . . 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 8849 . . . . . . . . . . 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 8852 . . . . . . . . . 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 2318 . . . . . . . . 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 734 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  c )  =  0 )
3735, 36eqtr3d 2318 . . . . . . . 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 5827 . . . . . . . . . 10  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( a  +  ( _i  x.  b ) )  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
3938eqeq1d 2292 . . . . . . . . 9  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( ( a  +  ( _i  x.  b
) )  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) )  =  0 ) )
4039rspcev 2885 . . . . . . . 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 644 . . . . . . 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 425 . . . . . 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 2675 . . . . 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 16 . . . 4  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  E. x  e.  CC  ( ( a  +  ( _i  x.  b
) )  +  x
)  =  0 )
45 oveq1 5826 . . . . . 6  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  x ) )
4645eqeq1d 2292 . . . . 5  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4746rexbidv 2565 . . . 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 215 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( A  =  ( a  +  ( _i  x.  b ) )  ->  E. x  e.  CC  ( A  +  x
)  =  0 ) )
4948rexlimivv 2673 . 2  |-  ( E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b
) )  ->  E. x  e.  CC  ( A  +  x )  =  0 )
501, 49syl 17 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   E.wrex 2545  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   _ici 8734    + caddc 8735    x. cmul 8737
This theorem is referenced by:  addid2  8990  addcan2  8992  0cnALT  9036  negeu  9037
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-ltxr 8867
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