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Theorem cnaddabl 16340
Description: The complex numbers are an Abelian group under addition. This version of cnaddablx 16339 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 17813. (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 9355 . . . 4  |-  CC  e.  _V
2 cnaddabl.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  CC >. ,  <. ( +g  `  ndx ) ,  +  >. }
32grpbase 14270 . . . 4  |-  ( CC  e.  _V  ->  CC  =  ( Base `  G
) )
41, 3ax-mp 5 . . 3  |-  CC  =  ( Base `  G )
5 addex 10981 . . . 4  |-  +  e.  _V
62grpplusg 14271 . . . 4  |-  (  +  e.  _V  ->  +  =  ( +g  `  G
) )
75, 6ax-mp 5 . . 3  |-  +  =  ( +g  `  G )
8 addcl 9356 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
9 addass 9361 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
10 0cn 9370 . . 3  |-  0  e.  CC
11 addid2 9544 . . 3  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
12 negcl 9602 . . 3  |-  ( x  e.  CC  ->  -u x  e.  CC )
13 addcom 9547 . . . . 5  |-  ( ( x  e.  CC  /\  -u x  e.  CC )  ->  ( x  +  -u x )  =  (
-u x  +  x
) )
1412, 13mpdan 668 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  ( -u x  +  x )
)
15 negid 9648 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  0 )
1614, 15eqtr3d 2472 . . 3  |-  ( x  e.  CC  ->  ( -u x  +  x )  =  0 )
174, 7, 8, 9, 10, 11, 12, 16isgrpi 15555 . 2  |-  G  e. 
Grp
18 addcom 9547 . 2  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
1917, 4, 7, 18isabli 16282 1  |-  G  e. 
Abel
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   _Vcvv 2967   {cpr 3874   <.cop 3878   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274    + caddc 9277   -ucneg 9588   ndxcnx 14163   Basecbs 14166   +g cplusg 14230   Abelcabel 16269
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-addf 9353
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15536  df-cmn 16270  df-abl 16271
This theorem is referenced by:  cnaddcom  32457
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