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Theorem cnid 26072
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 26071 . . . 4  |-  +  e.  AbelOp
2 ablogrpo 26005 . . . 4  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
31, 2ax-mp 5 . . 3  |-  +  e.  GrpOp
4 ax-addf 9615 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
54fdmi 5732 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
63, 5grporn 25933 . . . 4  |-  CC  =  ran  +
7 eqid 2450 . . . 4  |-  (GId `  +  )  =  (GId `  +  )
86, 7grpoidval 25937 . . 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 9813 . . . 4  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
1110rgen 2746 . . 3  |-  A. x  e.  CC  ( 0  +  x )  =  x
12 0cn 9632 . . . 4  |-  0  e.  CC
136grpoideu 25930 . . . . 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 6295 . . . . . . 7  |-  ( y  =  0  ->  (
y  +  x )  =  ( 0  +  x ) )
1615eqeq1d 2452 . . . . . 6  |-  ( y  =  0  ->  (
( y  +  x
)  =  x  <->  ( 0  +  x )  =  x ) )
1716ralbidv 2826 . . . . 5  |-  ( y  =  0  ->  ( A. x  e.  CC  ( y  +  x
)  =  x  <->  A. x  e.  CC  ( 0  +  x )  =  x ) )
1817riota2 6272 . . . 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 677 . . 3  |-  ( A. x  e.  CC  (
0  +  x )  =  x  <->  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )  =  0 )
2011, 19mpbi 212 . 2  |-  ( iota_ y  e.  CC  A. x  e.  CC  ( y  +  x )  =  x )  =  0
219, 20eqtr2i 2473 1  |-  0  =  (GId `  +  )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1443    e. wcel 1886   A.wral 2736   E!wreu 2738    X. cxp 4831   ` cfv 5581   iota_crio 6249  (class class class)co 6288   CCcc 9534   0cc0 9536    + caddc 9539   GrpOpcgr 25907  GIdcgi 25908   AbelOpcablo 26002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-addf 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-ltxr 9677  df-sub 9859  df-neg 9860  df-grpo 25912  df-gid 25913  df-ablo 26003
This theorem is referenced by:  addinv  26073  readdsubgo  26074  zaddsubgo  26075  cnnv  26301
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