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Theorem cnid 25057
Description: The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnid  |-  0  =  (GId `  +  )

Proof of Theorem cnid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnaddablo 25056 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 24990 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9571 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5736 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 24918 . . . 4  |-  CC  =  ran  +
7 eqid 2467 . . . 4  |-  (GId `  +  )  =  (GId `  +  )
86, 7grpoidval 24922 . . 3  |-  (  +  e.  GrpOp  ->  (GId `  +  )  =  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x ) )
93, 8ax-mp 5 . 2  |-  (GId `  +  )  =  ( iota_ y  e.  CC  A. x  e.  CC  (
y  +  x )  =  x )
10 addid2 9762 . . . 4  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
1110rgen 2824 . . 3  |-  A. x  e.  CC  ( 0  +  x )  =  x
12 0cn 9588 . . . 4  |-  0  e.  CC
136grpoideu 24915 . . . . 5  |-  (  +  e.  GrpOp  ->  E! y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )
143, 13ax-mp 5 . . . 4  |-  E! y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x
15 oveq1 6291 . . . . . . 7  |-  ( y  =  0  ->  (
y  +  x )  =  ( 0  +  x ) )
1615eqeq1d 2469 . . . . . 6  |-  ( y  =  0  ->  (
( y  +  x
)  =  x  <->  ( 0  +  x )  =  x ) )
1716ralbidv 2903 . . . . 5  |-  ( y  =  0  ->  ( A. x  e.  CC  ( y  +  x
)  =  x  <->  A. x  e.  CC  ( 0  +  x )  =  x ) )
1817riota2 6268 . . . 4  |-  ( ( 0  e.  CC  /\  E! y  e.  CC  A. x  e.  CC  (
y  +  x )  =  x )  -> 
( A. x  e.  CC  ( 0  +  x )  =  x  <-> 
( iota_ y  e.  CC  A. x  e.  CC  (
y  +  x )  =  x )  =  0 ) )
1912, 14, 18mp2an 672 . . 3  |-  ( A. x  e.  CC  (
0  +  x )  =  x  <->  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )  =  0 )
2011, 19mpbi 208 . 2  |-  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )  =  0
219, 20eqtr2i 2497 1  |-  0  =  (GId `  +  )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2814   E!wreu 2816    X. cxp 4997   ` cfv 5588   iota_crio 6244  (class class class)co 6284   CCcc 9490   0cc0 9492    + caddc 9495   GrpOpcgr 24892  GIdcgi 24893   AbelOpcablo 24987
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-addf 9571
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807  df-neg 9808  df-grpo 24897  df-gid 24898  df-ablo 24988
This theorem is referenced by:  addinv  25058  readdsubgo  25059  zaddsubgo  25060  cnnv  25286
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