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Theorem cnaddabl 15445
Description: The complex numbers are an Abelian group under addition. This version of cnaddablx 15444 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how  Base and  +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnrng 16686. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
Hypothesis
Ref Expression
cnaddabl.g  |-  G  =  { <. ( Base `  ndx ) ,  CC >. ,  <. ( +g  `  ndx ) ,  +  >. }
Assertion
Ref Expression
cnaddabl  |-  G  e. 
Abel

Proof of Theorem cnaddabl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 9035 . . . 4  |-  CC  e.  _V
2 cnaddabl.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  CC >. ,  <. ( +g  `  ndx ) ,  +  >. }
32grpbase 13532 . . . 4  |-  ( CC  e.  _V  ->  CC  =  ( Base `  G
) )
41, 3ax-mp 8 . . 3  |-  CC  =  ( Base `  G )
5 addex 10574 . . . 4  |-  +  e.  _V
62grpplusg 13533 . . . 4  |-  (  +  e.  _V  ->  +  =  ( +g  `  G
) )
75, 6ax-mp 8 . . 3  |-  +  =  ( +g  `  G )
8 addcl 9036 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
9 addass 9041 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
10 0cn 9048 . . 3  |-  0  e.  CC
11 addid2 9213 . . 3  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
12 negcl 9270 . . 3  |-  ( x  e.  CC  ->  -u x  e.  CC )
13 addcom 9216 . . . . 5  |-  ( ( x  e.  CC  /\  -u x  e.  CC )  ->  ( x  +  -u x )  =  (
-u x  +  x
) )
1412, 13mpdan 650 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  ( -u x  +  x )
)
15 negid 9312 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  0 )
1614, 15eqtr3d 2446 . . 3  |-  ( x  e.  CC  ->  ( -u x  +  x )  =  0 )
174, 7, 8, 9, 10, 11, 12, 16isgrpi 14794 . 2  |-  G  e. 
Grp
18 addcom 9216 . 2  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
1917, 4, 7, 18isabli 15389 1  |-  G  e. 
Abel
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2924   {cpr 3783   <.cop 3785   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954    + caddc 8957   -ucneg 9256   ndxcnx 13429   Basecbs 13432   +g cplusg 13492   Abelcabel 15376
This theorem is referenced by:  cnaddcom  29466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-addf 9033
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-0g 13690  df-mnd 14653  df-grp 14775  df-cmn 15377  df-abl 15378
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