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Theorem cnrngo 21974
Description: The set of complex numbers is a (unital) ring. (Contributed by Steve Rodriguez, 2-Feb-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnrngo  |-  <.  +  ,  x.  >.  e.  RingOps

Proof of Theorem cnrngo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnaddablo 21921 . . 3  |-  +  e.  AbelOp
2 ax-mulf 9054 . . 3  |-  x.  :
( CC  X.  CC )
--> CC
31, 2pm3.2i 442 . 2  |-  (  +  e.  AbelOp  /\  x.  : ( CC  X.  CC ) --> CC )
4 mulass 9062 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
5 adddi 9063 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
6 adddir 9067 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
74, 5, 63jca 1134 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) )  /\  ( x  x.  (
y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z
)  =  ( ( x  x.  z )  +  ( y  x.  z ) ) ) )
87rgen3 2790 . . 3  |-  A. x  e.  CC  A. y  e.  CC  A. z  e.  CC  ( ( ( x  x.  y )  x.  z )  =  ( x  x.  (
y  x.  z ) )  /\  ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )
9 ax-1cn 9032 . . . 4  |-  1  e.  CC
10 mulid2 9073 . . . . . 6  |-  ( y  e.  CC  ->  (
1  x.  y )  =  y )
11 mulid1 9072 . . . . . 6  |-  ( y  e.  CC  ->  (
y  x.  1 )  =  y )
1210, 11jca 519 . . . . 5  |-  ( y  e.  CC  ->  (
( 1  x.  y
)  =  y  /\  ( y  x.  1 )  =  y ) )
1312rgen 2758 . . . 4  |-  A. y  e.  CC  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y )
14 oveq1 6074 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
1514eqeq1d 2438 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  =  y  <->  ( 1  x.  y )  =  y ) )
16 oveq2 6075 . . . . . . . 8  |-  ( x  =  1  ->  (
y  x.  x )  =  ( y  x.  1 ) )
1716eqeq1d 2438 . . . . . . 7  |-  ( x  =  1  ->  (
( y  x.  x
)  =  y  <->  ( y  x.  1 )  =  y ) )
1815, 17anbi12d 692 . . . . . 6  |-  ( x  =  1  ->  (
( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )  <->  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y ) ) )
1918ralbidv 2712 . . . . 5  |-  ( x  =  1  ->  ( A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )  <->  A. y  e.  CC  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y ) ) )
2019rspcev 3039 . . . 4  |-  ( ( 1  e.  CC  /\  A. y  e.  CC  (
( 1  x.  y
)  =  y  /\  ( y  x.  1 )  =  y ) )  ->  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y ) )
219, 13, 20mp2an 654 . . 3  |-  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )
228, 21pm3.2i 442 . 2  |-  ( A. x  e.  CC  A. y  e.  CC  A. z  e.  CC  ( ( ( x  x.  y )  x.  z )  =  ( x  x.  (
y  x.  z ) )  /\  ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )  /\  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y ) )
23 mulex 10595 . . 3  |-  x.  e.  _V
24 ablogrpo 21855 . . . . . 6  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
251, 24ax-mp 8 . . . . 5  |-  +  e.  GrpOp
26 ax-addf 9053 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
2726fdmi 5582 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
2825, 27grporn 21783 . . . 4  |-  CC  =  ran  +
2928isrngo 21949 . . 3  |-  (  x.  e.  _V  ->  ( <.  +  ,  x.  >.  e.  RingOps  <->  ( (  +  e.  AbelOp  /\  x.  : ( CC 
X.  CC ) --> CC )  /\  ( A. x  e.  CC  A. y  e.  CC  A. z  e.  CC  ( ( ( x  x.  y )  x.  z )  =  ( x  x.  (
y  x.  z ) )  /\  ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )  /\  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y ) ) ) ) )
3023, 29ax-mp 8 . 2  |-  ( <.  +  ,  x.  >.  e.  RingOps  <->  ( (  +  e.  AbelOp  /\  x.  : ( CC  X.  CC ) --> CC )  /\  ( A. x  e.  CC  A. y  e.  CC  A. z  e.  CC  (
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) )  /\  ( x  x.  (
y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z
)  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )  /\  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y ) ) ) )
313, 22, 30mpbir2an 887 1  |-  <.  +  ,  x.  >.  e.  RingOps
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2692   E.wrex 2693   _Vcvv 2943   <.cop 3804    X. cxp 4862   -->wf 5436  (class class class)co 6067   CCcc 8972   1c1 8975    + caddc 8977    x. cmul 8979   GrpOpcgr 21757   AbelOpcablo 21852   RingOpscrngo 21946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-addf 9053  ax-mulf 9054
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-po 4490  df-so 4491  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-riota 6535  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-ltxr 9109  df-sub 9277  df-neg 9278  df-grpo 21762  df-ablo 21853  df-rngo 21947
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