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Theorem cnegex2 8994
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 8796 . . . 4  |-  _i  e.  CC
21, 1mulcli 8842 . . 3  |-  ( _i  x.  _i )  e.  CC
3 mulcl 8821 . . 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 8836 . . . 4  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65oveq2d 5874 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
7 ax-i2m1 8805 . . . . 5  |-  ( ( _i  x.  _i )  +  1 )  =  0
87oveq1i 5868 . . . 4  |-  ( ( ( _i  x.  _i )  +  1 )  x.  A )  =  ( 0  x.  A
)
9 ax-1cn 8795 . . . . 5  |-  1  e.  CC
10 adddir 8830 . . . . 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 8990 . . . 4  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
138, 11, 123eqtr3a 2339 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  0 )
146, 13eqtr3d 2317 . 2  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 )
15 oveq1 5865 . . . 4  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
x  +  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
1615eqeq1d 2291 . . 3  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
( x  +  A
)  =  0  <->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 ) )
1716rspcev 2884 . 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 1623    e. wcel 1684   E.wrex 2544  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742
This theorem is referenced by:  addcan  8996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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