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Theorem cnegex 9772
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 9604 . 2  |-  ( A  e.  CC  ->  E. a  e.  RR  E. b  e.  RR  A  =  ( a  +  ( _i  x.  b ) ) )
2 ax-rnegex 9575 . . . . . . 7  |-  ( a  e.  RR  ->  E. c  e.  RR  ( a  +  c )  =  0 )
3 ax-rnegex 9575 . . . . . . 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 3034 . . . . . 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 9563 . . . . . . . . . . 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 760 . . . . . . . . . . 11  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  RR )
109recnd 9634 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  d  e.  CC )
118, 10mulcld 9628 . . . . . . . . 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 759 . . . . . . . . . 10  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  RR )
1312recnd 9634 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  c  e.  CC )
1411, 13addcld 9627 . . . . . . . 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 757 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  RR )
1615recnd 9634 . . . . . . . . . . . . 13  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  a  e.  CC )
17 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  RR )
1817recnd 9634 . . . . . . . . . . . . . 14  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  b  e.  CC )
198, 18mulcld 9628 . . . . . . . . . . . . 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 9630 . . . . . . . . . . . 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 9632 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( b  +  d )  =  0 )
2322oveq2d 6311 . . . . . . . . . . . . . . 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 9770 . . . . . . . . . . . . . . . 16  |-  ( _i  e.  CC  ->  (
_i  x.  0 )  =  0 )
257, 24ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( _i  x.  0 )  =  0
2623, 25syl6eq 2524 . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . 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 6311 . . . . . . . . . . . 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 9771 . . . . . . . . . . . . 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 2512 . . . . . . . . . . 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 6310 . . . . . . . . . 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 9627 . . . . . . . . . . 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 9630 . . . . . . . . . 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 2510 . . . . . . . . 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 755 . . . . . . . . 9  |-  ( ( ( ( a  e.  RR  /\  b  e.  RR )  /\  (
c  e.  RR  /\  d  e.  RR )
)  /\  ( (
a  +  c )  =  0  /\  (
b  +  d )  =  0 ) )  ->  ( a  +  c )  =  0 )
3735, 36eqtr3d 2510 . . . . . . . 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 6303 . . . . . . . . . 10  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( a  +  ( _i  x.  b ) )  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) ) )
3938eqeq1d 2469 . . . . . . . . 9  |-  ( x  =  ( ( _i  x.  d )  +  c )  ->  (
( ( a  +  ( _i  x.  b
) )  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  ( ( _i  x.  d )  +  c ) )  =  0 ) )
4039rspcev 3219 . . . . . . . 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 2966 . . . . 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 6302 . . . . . 6  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  ( A  +  x )  =  ( ( a  +  ( _i  x.  b ) )  +  x ) )
4645eqeq1d 2469 . . . . 5  |-  ( A  =  ( a  +  ( _i  x.  b
) )  ->  (
( A  +  x
)  =  0  <->  (
( a  +  ( _i  x.  b ) )  +  x )  =  0 ) )
4746rexbidv 2978 . . . 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 2964 . 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 1379    e. wcel 1767   E.wrex 2818  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   _ici 9506    + caddc 9507    x. cmul 9509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645
This theorem is referenced by:  addid2  9774  addcan2  9776  0cnALT  9821  negeu  9822
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