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Theorem cnegex 8873
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
StepHypRef Expression
1 cnre 8716 . 2  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 ax-rnegex 8688 . . . . . . 7  |-  ( a  e.  RR  ->  E. c  e.  RR  ( a  +  c )  =  0 )
3 ax-rnegex 8688 . . . . . . 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 2669 . . . . . 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 8676 . . . . . . . . . . 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 740 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  RR )
109recnd 8741 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  CC )
118, 10mulcld 8735 . . . . . . . . 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 739 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  RR )
1312recnd 8741 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  CC )
1411, 13addcld 8734 . . . . . . . 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 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  RR )
1615recnd 8741 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  CC )
17 simpllr 738 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  RR )
1817recnd 8741 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  CC )
198, 18mulcld 8735 . . . . . . . . . . . . 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 8737 . . . . . . . . . . . 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 8739 . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( b  +  d )  =  0 )
2322oveq2d 5726 . . . . . . . . . . . . . . 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 8871 . . . . . . . . . . . . . . . 16  |-  ( _i  e.  CC  ->  (
_i  x.  0 )  =  0 )
257, 24ax-mp 10 . . . . . . . . . . . . . . 15  |-  ( _i  x.  0 )  =  0
2623, 25syl6eq 2301 . . . . . . . . . . . . . 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 2287 . . . . . . . . . . . . 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 5726 . . . . . . . . . . . 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 8872 . . . . . . . . . . . . 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 2289 . . . . . . . . . . 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 5725 . . . . . . . . . 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 8734 . . . . . . . . . . 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 8737 . . . . . . . . . 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 2287 . . . . . . . . 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 735 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  c )  =  0 )
3735, 36eqtr3d 2287 . . . . . . . 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 5718 . . . . . . . . . 10  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( a  +  ( _i  x.  b ) )  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
3938eqeq1d 2261 . . . . . . . . 9  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( ( a  +  ( _i  x.  b
) )  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) )  =  0 ) )
4039rcla4ev 2821 . . . . . . . 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 645 . . . . . . 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 2636 . . . . 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 5717 . . . . . 6  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  x ) )
4645eqeq1d 2261 . . . . 5  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4746rexbidv 2528 . . . 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 2634 . 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 1619    e. wcel 1621   E.wrex 2510  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   _ici 8619    + caddc 8620    x. cmul 8622
This theorem is referenced by:  addid2  8875  addcan2  8877  0cnALT  8921  negeu  8922
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752
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