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Theorem addinv 21942
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 21940 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 21874 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 8 . . 3  |-  +  e.  GrpOp
4 ax-addf 9071 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5598 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 21802 . . . 4  |-  CC  =  ran  +
7 cnid 21941 . . . 4  |-  0  =  (GId `  +  )
8 eqid 2438 . . . 4  |-  ( inv `  +  )  =  ( inv `  +  )
96, 7, 8grpoinvval 21815 . . 3  |-  ( (  +  e.  GrpOp  /\  A  e.  CC )  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC ( y  +  A )  =  0 ) )
103, 9mpan 653 . 2  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC ( y  +  A )  =  0 ) )
11 df-neg 9296 . . . . 5  |-  -u A  =  ( 0  -  A )
1211oveq1i 6093 . . . 4  |-  ( -u A  +  A )  =  ( ( 0  -  A )  +  A )
13 0cn 9086 . . . . 5  |-  0  e.  CC
14 npcan 9316 . . . . 5  |-  ( ( 0  e.  CC  /\  A  e.  CC )  ->  ( ( 0  -  A )  +  A
)  =  0 )
1513, 14mpan 653 . . . 4  |-  ( A  e.  CC  ->  (
( 0  -  A
)  +  A )  =  0 )
1612, 15syl5eq 2482 . . 3  |-  ( A  e.  CC  ->  ( -u A  +  A )  =  0 )
17 negcl 9308 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
18 negeu 9298 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  E! y  e.  CC  ( A  +  y
)  =  0 )
1913, 18mpan2 654 . . . . 5  |-  ( A  e.  CC  ->  E! y  e.  CC  ( A  +  y )  =  0 )
20 addcom 9254 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  +  y )  =  ( y  +  A ) )
2120eqeq1d 2446 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  <-> 
( y  +  A
)  =  0 ) )
2221reubidva 2893 . . . . 5  |-  ( A  e.  CC  ->  ( E! y  e.  CC  ( A  +  y
)  =  0  <->  E! y  e.  CC  (
y  +  A )  =  0 ) )
2319, 22mpbid 203 . . . 4  |-  ( A  e.  CC  ->  E! y  e.  CC  (
y  +  A )  =  0 )
24 oveq1 6090 . . . . . 6  |-  ( y  =  -u A  ->  (
y  +  A )  =  ( -u A  +  A ) )
2524eqeq1d 2446 . . . . 5  |-  ( y  =  -u A  ->  (
( y  +  A
)  =  0  <->  ( -u A  +  A )  =  0 ) )
2625riota2 6574 . . . 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 644 . . 3  |-  ( A  e.  CC  ->  (
( -u A  +  A
)  =  0  <->  ( iota_ y  e.  CC ( y  +  A )  =  0 )  = 
-u A ) )
2816, 27mpbid 203 . 2  |-  ( A  e.  CC  ->  ( iota_ y  e.  CC ( y  +  A )  =  0 )  = 
-u A )
2910, 28eqtrd 2470 1  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!wreu 2709    X. cxp 4878   ` cfv 5456  (class class class)co 6083   iota_crio 6544   CCcc 8990   0cc0 8992    + caddc 8995    - cmin 9293   -ucneg 9294   GrpOpcgr 21776   invcgn 21778   AbelOpcablo 21871
This theorem is referenced by:  readdsubgo  21943  zaddsubgo  21944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-addf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-ltxr 9127  df-sub 9295  df-neg 9296  df-grpo 21781  df-gid 21782  df-ginv 21783  df-ablo 21872
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