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Theorem cnegex2 9010
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 8812 . . . 4  |-  _i  e.  CC
21, 1mulcli 8858 . . 3  |-  ( _i  x.  _i )  e.  CC
3 mulcl 8837 . . 3  |-  ( ( ( _i  x.  _i )  e.  CC  /\  A  e.  CC )  ->  (
( _i  x.  _i )  x.  A )  e.  CC )
42, 3mpan 651 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  _i )  x.  A )  e.  CC )
5 mulid2 8852 . . . 4  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65oveq2d 5890 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
7 ax-i2m1 8821 . . . . 5  |-  ( ( _i  x.  _i )  +  1 )  =  0
87oveq1i 5884 . . . 4  |-  ( ( ( _i  x.  _i )  +  1 )  x.  A )  =  ( 0  x.  A
)
9 ax-1cn 8811 . . . . 5  |-  1  e.  CC
10 adddir 8846 . . . . 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 1267 . . . 4  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  +  1
)  x.  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  ( 1  x.  A
) ) )
12 mul02 9006 . . . 4  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
138, 11, 123eqtr3a 2352 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  0 )
146, 13eqtr3d 2330 . 2  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 )
15 oveq1 5881 . . . 4  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
x  +  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
1615eqeq1d 2304 . . 3  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
( x  +  A
)  =  0  <->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 ) )
1716rspcev 2897 . 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 642 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   E.wrex 2557  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758
This theorem is referenced by:  addcan  9012
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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