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Theorem cnrngo 22029
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 21976 . . 3  |-  +  e.  AbelOp
2 ax-mulf 9108 . . 3  |-  x.  :
( CC  X.  CC )
--> CC
31, 2pm3.2i 443 . 2  |-  (  +  e.  AbelOp  /\  x.  : ( CC  X.  CC ) --> CC )
4 mulass 9116 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
5 adddi 9117 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
6 adddir 9121 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
74, 5, 63jca 1135 . . . 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 2810 . . 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 9086 . . . 4  |-  1  e.  CC
10 mulid2 9127 . . . . . 6  |-  ( y  e.  CC  ->  (
1  x.  y )  =  y )
11 mulid1 9126 . . . . . 6  |-  ( y  e.  CC  ->  (
y  x.  1 )  =  y )
1210, 11jca 520 . . . . 5  |-  ( y  e.  CC  ->  (
( 1  x.  y
)  =  y  /\  ( y  x.  1 )  =  y ) )
1312rgen 2778 . . . 4  |-  A. y  e.  CC  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y )
14 oveq1 6124 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
1514eqeq1d 2451 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  =  y  <->  ( 1  x.  y )  =  y ) )
16 oveq2 6125 . . . . . . . 8  |-  ( x  =  1  ->  (
y  x.  x )  =  ( y  x.  1 ) )
1716eqeq1d 2451 . . . . . . 7  |-  ( x  =  1  ->  (
( y  x.  x
)  =  y  <->  ( y  x.  1 )  =  y ) )
1815, 17anbi12d 693 . . . . . 6  |-  ( x  =  1  ->  (
( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )  <->  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y ) ) )
1918ralbidv 2732 . . . . 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 3061 . . . 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 655 . . 3  |-  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )
228, 21pm3.2i 443 . 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 10649 . . 3  |-  x.  e.  _V
24 ablogrpo 21910 . . . . . 6  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
251, 24ax-mp 5 . . . . 5  |-  +  e.  GrpOp
26 ax-addf 9107 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
2726fdmi 5631 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
2825, 27grporn 21838 . . . 4  |-  CC  =  ran  +
2928isrngo 22004 . . 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 5 . 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 888 1  |-  <.  +  ,  x.  >.  e.  RingOps
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   A.wral 2712   E.wrex 2713   _Vcvv 2965   <.cop 3846    X. cxp 4911   -->wf 5485  (class class class)co 6117   CCcc 9026   1c1 9029    + caddc 9031    x. cmul 9033   GrpOpcgr 21812   AbelOpcablo 21907   RingOpscrngo 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-addf 9107  ax-mulf 9108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-po 4538  df-so 4539  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-ltxr 9163  df-sub 9331  df-neg 9332  df-grpo 21817  df-ablo 21908  df-rngo 22002
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