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Theorem cnegex2 9248
Description: Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
cnegex2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
Distinct variable group:    x, A

Proof of Theorem cnegex2
StepHypRef Expression
1 ax-icn 9049 . . . 4  |-  _i  e.  CC
21, 1mulcli 9095 . . 3  |-  ( _i  x.  _i )  e.  CC
3 mulcl 9074 . . 3  |-  ( ( ( _i  x.  _i )  e.  CC  /\  A  e.  CC )  ->  (
( _i  x.  _i )  x.  A )  e.  CC )
42, 3mpan 652 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  _i )  x.  A )  e.  CC )
5 mulid2 9089 . . . 4  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65oveq2d 6097 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
7 ax-i2m1 9058 . . . . 5  |-  ( ( _i  x.  _i )  +  1 )  =  0
87oveq1i 6091 . . . 4  |-  ( ( ( _i  x.  _i )  +  1 )  x.  A )  =  ( 0  x.  A
)
9 ax-1cn 9048 . . . . 5  |-  1  e.  CC
10 adddir 9083 . . . . 5  |-  ( ( ( _i  x.  _i )  e.  CC  /\  1  e.  CC  /\  A  e.  CC )  ->  (
( ( _i  x.  _i )  +  1
)  x.  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  ( 1  x.  A
) ) )
112, 9, 10mp3an12 1269 . . . 4  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  +  1
)  x.  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  ( 1  x.  A
) ) )
12 mul02 9244 . . . 4  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
138, 11, 123eqtr3a 2492 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  0 )
146, 13eqtr3d 2470 . 2  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 )
15 oveq1 6088 . . . 4  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
x  +  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
1615eqeq1d 2444 . . 3  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
( x  +  A
)  =  0  <->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 ) )
1716rspcev 3052 . 2  |-  ( ( ( ( _i  x.  _i )  x.  A
)  e.  CC  /\  ( ( ( _i  x.  _i )  x.  A )  +  A
)  =  0 )  ->  E. x  e.  CC  ( x  +  A
)  =  0 )
184, 14, 17syl2anc 643 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2706  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995
This theorem is referenced by:  addcan  9250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125
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