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Theorem cnid 25950
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 25949 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 25883 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9607 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5742 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 25811 . . . 4  |-  CC  =  ran  +
7 eqid 2420 . . . 4  |-  (GId `  +  )  =  (GId `  +  )
86, 7grpoidval 25815 . . 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 9805 . . . 4  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
1110rgen 2783 . . 3  |-  A. x  e.  CC  ( 0  +  x )  =  x
12 0cn 9624 . . . 4  |-  0  e.  CC
136grpoideu 25808 . . . . 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 6303 . . . . . . 7  |-  ( y  =  0  ->  (
y  +  x )  =  ( 0  +  x ) )
1615eqeq1d 2422 . . . . . 6  |-  ( y  =  0  ->  (
( y  +  x
)  =  x  <->  ( 0  +  x )  =  x ) )
1716ralbidv 2862 . . . . 5  |-  ( y  =  0  ->  ( A. x  e.  CC  ( y  +  x
)  =  x  <->  A. x  e.  CC  ( 0  +  x )  =  x ) )
1817riota2 6280 . . . 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 676 . . 3  |-  ( A. x  e.  CC  (
0  +  x )  =  x  <->  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )  =  0 )
2011, 19mpbi 211 . 2  |-  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )  =  0
219, 20eqtr2i 2450 1  |-  0  =  (GId `  +  )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1867   A.wral 2773   E!wreu 2775    X. cxp 4843   ` cfv 5592   iota_crio 6257  (class class class)co 6296   CCcc 9526   0cc0 9528    + caddc 9531   GrpOpcgr 25785  GIdcgi 25786   AbelOpcablo 25880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-addf 9607
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-ltxr 9669  df-sub 9851  df-neg 9852  df-grpo 25790  df-gid 25791  df-ablo 25881
This theorem is referenced by:  addinv  25951  readdsubgo  25952  zaddsubgo  25953  cnnv  26179
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