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Theorem cnaddabl 15258
Description: The complex numbers are an Abelian group under addition. This version of cnaddablx 15257 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 16502. (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 8908 . . . 4  |-  CC  e.  _V
2 cnaddabl.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  CC >. ,  <. ( +g  `  ndx ) ,  +  >. }
32grpbase 13345 . . . 4  |-  ( CC  e.  _V  ->  CC  =  ( Base `  G
) )
41, 3ax-mp 8 . . 3  |-  CC  =  ( Base `  G )
5 addex 10444 . . . 4  |-  +  e.  _V
62grpplusg 13346 . . . 4  |-  (  +  e.  _V  ->  +  =  ( +g  `  G
) )
75, 6ax-mp 8 . . 3  |-  +  =  ( +g  `  G )
8 addcl 8909 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
9 addass 8914 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  +  z )  =  ( x  +  ( y  +  z ) ) )
10 0cn 8921 . . 3  |-  0  e.  CC
11 addid2 9085 . . 3  |-  ( x  e.  CC  ->  (
0  +  x )  =  x )
12 negcl 9142 . . 3  |-  ( x  e.  CC  ->  -u x  e.  CC )
13 addcom 9088 . . . . 5  |-  ( ( x  e.  CC  /\  -u x  e.  CC )  ->  ( x  +  -u x )  =  (
-u x  +  x
) )
1412, 13mpdan 649 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  ( -u x  +  x )
)
15 negid 9184 . . . 4  |-  ( x  e.  CC  ->  (
x  +  -u x
)  =  0 )
1614, 15eqtr3d 2392 . . 3  |-  ( x  e.  CC  ->  ( -u x  +  x )  =  0 )
174, 7, 8, 9, 10, 11, 12, 16isgrpi 14607 . 2  |-  G  e. 
Grp
18 addcom 9088 . 2  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
1917, 4, 7, 18isabli 15202 1  |-  G  e. 
Abel
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   _Vcvv 2864   {cpr 3717   <.cop 3719   ` cfv 5337  (class class class)co 5945   CCcc 8825   0cc0 8827    + caddc 8830   -ucneg 9128   ndxcnx 13242   Basecbs 13245   +g cplusg 13305   Abelcabel 15189
This theorem is referenced by:  cnaddcom  29230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-addf 8906
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-plusg 13318  df-0g 13503  df-mnd 14466  df-grp 14588  df-cmn 15190  df-abl 15191
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