MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addinv Structured version   Unicode version

Theorem addinv 25046
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 25044 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 24978 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9570 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5735 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 24906 . . . 4  |-  CC  =  ran  +
7 cnid 25045 . . . 4  |-  0  =  (GId `  +  )
8 eqid 2467 . . . 4  |-  ( inv `  +  )  =  ( inv `  +  )
96, 7, 8grpoinvval 24919 . . 3  |-  ( (  +  e.  GrpOp  /\  A  e.  CC )  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
103, 9mpan 670 . 2  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  ( iota_ y  e.  CC  ( y  +  A )  =  0 ) )
11 df-neg 9807 . . . . 5  |-  -u A  =  ( 0  -  A )
1211oveq1i 6293 . . . 4  |-  ( -u A  +  A )  =  ( ( 0  -  A )  +  A )
13 0cn 9587 . . . . 5  |-  0  e.  CC
14 npcan 9828 . . . . 5  |-  ( ( 0  e.  CC  /\  A  e.  CC )  ->  ( ( 0  -  A )  +  A
)  =  0 )
1513, 14mpan 670 . . . 4  |-  ( A  e.  CC  ->  (
( 0  -  A
)  +  A )  =  0 )
1612, 15syl5eq 2520 . . 3  |-  ( A  e.  CC  ->  ( -u A  +  A )  =  0 )
17 negcl 9819 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
18 negeu 9809 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  E! y  e.  CC  ( A  +  y
)  =  0 )
1913, 18mpan2 671 . . . . 5  |-  ( A  e.  CC  ->  E! y  e.  CC  ( A  +  y )  =  0 )
20 addcom 9764 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  +  y )  =  ( y  +  A ) )
2120eqeq1d 2469 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( A  +  y )  =  0  <-> 
( y  +  A
)  =  0 ) )
2221reubidva 3045 . . . . 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 6290 . . . . . 6  |-  ( y  =  -u A  ->  (
y  +  A )  =  ( -u A  +  A ) )
2524eqeq1d 2469 . . . . 5  |-  ( y  =  -u A  ->  (
( y  +  A
)  =  0  <->  ( -u A  +  A )  =  0 ) )
2625riota2 6267 . . . 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 661 . . 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 2508 1  |-  ( A  e.  CC  ->  (
( inv `  +  ) `  A )  =  -u A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E!wreu 2816    X. cxp 4997   ` cfv 5587   iota_crio 6243  (class class class)co 6283   CCcc 9489   0cc0 9491    + caddc 9494    - cmin 9804   -ucneg 9805   GrpOpcgr 24880   invcgn 24882   AbelOpcablo 24975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-addf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-ltxr 9632  df-sub 9806  df-neg 9807  df-grpo 24885  df-gid 24886  df-ginv 24887  df-ablo 24976
This theorem is referenced by:  readdsubgo  25047  zaddsubgo  25048
  Copyright terms: Public domain W3C validator