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Theorem addinv 25552
Description: Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addinv  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )

Proof of Theorem addinv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnaddablo 25550 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 25484 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9560 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5718 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 25412 . . . 4  |-  CC  =  ran  +
7 cnid 25551 . . . 4  |-  0  =  (GId `  +  )
8 eqid 2454 . . . 4  |-  ( inv `  +  )  =  ( inv `  +  )
96, 7, 8grpoinvval 25425 . . 3  |-  ( (  +  e.  GrpOp  /\  A  e.  CC )  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
103, 9mpan 668 . 2  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
11 df-neg 9799 . . . . 5  |-  -u A  =  ( 0  -  A )
1211oveq1i 6280 . . . 4  |-  ( -u A  +  A )  =  ( ( 0  -  A )  +  A )
13 0cn 9577 . . . . 5  |-  0  e.  CC
14 npcan 9820 . . . . 5  |-  ( ( 0  e.  CC  /\  A  e.  CC )  ->  ( ( 0  -  A )  +  A
)  =  0 )
1513, 14mpan 668 . . . 4  |-  ( A  e.  CC  ->  (
( 0  -  A
)  +  A )  =  0 )
1612, 15syl5eq 2507 . . 3  |-  ( A  e.  CC  ->  ( -u A  +  A )  =  0 )
17 negcl 9811 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
18 negeu 9801 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  E! y  e.  CC  ( A  +  y
)  =  0 )
1913, 18mpan2 669 . . . . 5  |-  ( A  e.  CC  ->  E! y  e.  CC  ( A  +  y )  =  0 )
20 addcom 9755 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  +  y )  =  ( y  +  A ) )
2120eqeq1d 2456 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  <-> 
( y  +  A
)  =  0 ) )
2221reubidva 3038 . . . . 5  |-  ( A  e.  CC  ->  ( E! y  e.  CC  ( A  +  y
)  =  0  <->  E! y  e.  CC  (
y  +  A )  =  0 ) )
2319, 22mpbid 210 . . . 4  |-  ( A  e.  CC  ->  E! y  e.  CC  (
y  +  A )  =  0 )
24 oveq1 6277 . . . . . 6  |-  ( y  =  -u A  ->  (
y  +  A )  =  ( -u A  +  A ) )
2524eqeq1d 2456 . . . . 5  |-  ( y  =  -u A  ->  (
( y  +  A
)  =  0  <->  ( -u A  +  A )  =  0 ) )
2625riota2 6254 . . . 4  |-  ( (
-u A  e.  CC  /\  E! y  e.  CC  ( y  +  A
)  =  0 )  ->  ( ( -u A  +  A )  =  0  <->  ( iota_ y  e.  CC  ( y  +  A )  =  0 )  =  -u A ) )
2717, 23, 26syl2anc 659 . . 3  |-  ( A  e.  CC  ->  (
( -u A  +  A
)  =  0  <->  ( iota_ y  e.  CC  (
y  +  A )  =  0 )  = 
-u A ) )
2816, 27mpbid 210 . 2  |-  ( A  e.  CC  ->  ( iota_ y  e.  CC  (
y  +  A )  =  0 )  = 
-u A )
2910, 28eqtrd 2495 1  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806    X. cxp 4986   ` cfv 5570   iota_crio 6231  (class class class)co 6270   CCcc 9479   0cc0 9481    + caddc 9484    - cmin 9796   -ucneg 9797   GrpOpcgr 25386   invcgn 25388   AbelOpcablo 25481
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-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  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  ax-addf 9560
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-reu 2811  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-iun 4317  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-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-grpo 25391  df-gid 25392  df-ginv 25393  df-ablo 25482
This theorem is referenced by:  readdsubgo  25553  zaddsubgo  25554
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