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Theorem cnid 23843
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 23842 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 23776 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9366 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5569 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 23704 . . . 4  |-  CC  =  ran  +
7 eqid 2443 . . . 4  |-  (GId `  +  )  =  (GId `  +  )
86, 7grpoidval 23708 . . 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 9557 . . . 4  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
1110rgen 2786 . . 3  |-  A. x  e.  CC  ( 0  +  x )  =  x
12 0cn 9383 . . . 4  |-  0  e.  CC
136grpoideu 23701 . . . . 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 6103 . . . . . . 7  |-  ( y  =  0  ->  (
y  +  x )  =  ( 0  +  x ) )
1615eqeq1d 2451 . . . . . 6  |-  ( y  =  0  ->  (
( y  +  x
)  =  x  <->  ( 0  +  x )  =  x ) )
1716ralbidv 2740 . . . . 5  |-  ( y  =  0  ->  ( A. x  e.  CC  ( y  +  x
)  =  x  <->  A. x  e.  CC  ( 0  +  x )  =  x ) )
1817riota2 6080 . . . 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 2464 1  |-  0  =  (GId `  +  )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2720   E!wreu 2722    X. cxp 4843   ` cfv 5423   iota_crio 6056  (class class class)co 6096   CCcc 9285   0cc0 9287    + caddc 9290   GrpOpcgr 23678  GIdcgi 23679   AbelOpcablo 23773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-addf 9366
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-sub 9602  df-neg 9603  df-grpo 23683  df-gid 23684  df-ablo 23774
This theorem is referenced by:  addinv  23844  readdsubgo  23845  zaddsubgo  23846  cnnv  24072
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