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Theorem cnaddabl 16473
Description: The complex numbers are an Abelian group under addition. This version of cnaddablx 16472 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 17966. (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 9477 . . . 4  |-  CC  e.  _V
2 cnaddabl.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  CC >. ,  <. ( +g  `  ndx ) ,  +  >. }
32grpbase 14400 . . . 4  |-  ( CC  e.  _V  ->  CC  =  ( Base `  G
) )
41, 3ax-mp 5 . . 3  |-  CC  =  ( Base `  G )
5 addex 11103 . . . 4  |-  +  e.  _V
62grpplusg 14401 . . . 4  |-  (  +  e.  _V  ->  +  =  ( +g  `  G
) )
75, 6ax-mp 5 . . 3  |-  +  =  ( +g  `  G )
8 addcl 9478 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
9 addass 9483 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
10 0cn 9492 . . 3  |-  0  e.  CC
11 addid2 9666 . . 3  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
12 negcl 9724 . . 3  |-  ( x  e.  CC  ->  -u x  e.  CC )
13 addcom 9669 . . . . 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 9770 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  0 )
1614, 15eqtr3d 2497 . . 3  |-  ( x  e.  CC  ->  ( -u x  +  x )  =  0 )
174, 7, 8, 9, 10, 11, 12, 16isgrpi 15686 . 2  |-  G  e. 
Grp
18 addcom 9669 . 2  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
1917, 4, 7, 18isabli 16415 1  |-  G  e. 
Abel
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3078   {cpr 3990   <.cop 3994   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396    + caddc 9399   -ucneg 9710   ndxcnx 14292   Basecbs 14295   +g cplusg 14360   Abelcabel 16402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-addf 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-plusg 14373  df-0g 14502  df-mnd 15537  df-grp 15667  df-cmn 16403  df-abl 16404
This theorem is referenced by:  cnaddcom  32975
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