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Theorem cnegex2 9751
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 9540 . . . 4  |-  _i  e.  CC
21, 1mulcli 9590 . . 3  |-  ( _i  x.  _i )  e.  CC
3 mulcl 9565 . . 3  |-  ( ( ( _i  x.  _i )  e.  CC  /\  A  e.  CC )  ->  (
( _i  x.  _i )  x.  A )  e.  CC )
42, 3mpan 668 . 2  |-  ( A  e.  CC  ->  (
( _i  x.  _i )  x.  A )  e.  CC )
5 mulid2 9583 . . . 4  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65oveq2d 6286 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
7 ax-i2m1 9549 . . . . 5  |-  ( ( _i  x.  _i )  +  1 )  =  0
87oveq1i 6280 . . . 4  |-  ( ( ( _i  x.  _i )  +  1 )  x.  A )  =  ( 0  x.  A
)
9 ax-1cn 9539 . . . . 5  |-  1  e.  CC
10 adddir 9576 . . . . 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 1312 . . . 4  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  +  1
)  x.  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  ( 1  x.  A
) ) )
12 mul02 9747 . . . 4  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
138, 11, 123eqtr3a 2519 . . 3  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  ( 1  x.  A ) )  =  0 )
146, 13eqtr3d 2497 . 2  |-  ( A  e.  CC  ->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 )
15 oveq1 6277 . . . 4  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
x  +  A )  =  ( ( ( _i  x.  _i )  x.  A )  +  A ) )
1615eqeq1d 2456 . . 3  |-  ( x  =  ( ( _i  x.  _i )  x.  A )  ->  (
( x  +  A
)  =  0  <->  (
( ( _i  x.  _i )  x.  A
)  +  A )  =  0 ) )
1716rspcev 3207 . 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 659 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   E.wrex 2805  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622
This theorem is referenced by:  addcan  9753
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