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Theorem cnrngo 21064
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 21011 . . 3  |-  +  e.  AbelOp
2 ax-mulf 8813 . . 3  |-  x.  :
( CC  X.  CC )
--> CC
31, 2pm3.2i 441 . 2  |-  (  +  e.  AbelOp  /\  x.  : ( CC  X.  CC ) --> CC )
4 mulass 8821 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
5 adddi 8822 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
6 adddir 8826 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
74, 5, 63jca 1132 . . . 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 2641 . . 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 8791 . . . 4  |-  1  e.  CC
10 mulid2 8832 . . . . . 6  |-  ( y  e.  CC  ->  (
1  x.  y )  =  y )
11 mulid1 8831 . . . . . 6  |-  ( y  e.  CC  ->  (
y  x.  1 )  =  y )
1210, 11jca 518 . . . . 5  |-  ( y  e.  CC  ->  (
( 1  x.  y
)  =  y  /\  ( y  x.  1 )  =  y ) )
1312rgen 2609 . . . 4  |-  A. y  e.  CC  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y )
14 oveq1 5827 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
1514eqeq1d 2292 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  =  y  <->  ( 1  x.  y )  =  y ) )
16 oveq2 5828 . . . . . . . 8  |-  ( x  =  1  ->  (
y  x.  x )  =  ( y  x.  1 ) )
1716eqeq1d 2292 . . . . . . 7  |-  ( x  =  1  ->  (
( y  x.  x
)  =  y  <->  ( y  x.  1 )  =  y ) )
1815, 17anbi12d 691 . . . . . 6  |-  ( x  =  1  ->  (
( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )  <->  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y ) ) )
1918ralbidv 2564 . . . . 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 2885 . . . 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 653 . . 3  |-  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )
228, 21pm3.2i 441 . 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 10349 . . 3  |-  x.  e.  _V
24 ablogrpo 20945 . . . . . 6  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
251, 24ax-mp 8 . . . . 5  |-  +  e.  GrpOp
26 ax-addf 8812 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
2726fdmi 5360 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
2825, 27grporn 20873 . . . 4  |-  CC  =  ran  +
2928isrngo 21039 . . 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 886 1  |-  <.  +  ,  x.  >.  e.  RingOps
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   A.wral 2544   E.wrex 2545   _Vcvv 2789   <.cop 3644    X. cxp 4686   -->wf 5217  (class class class)co 5820   CCcc 8731   1c1 8734    + caddc 8736    x. cmul 8738   GrpOpcgr 20847   AbelOpcablo 20942   RingOpscrngo 21036
This theorem is referenced by:  zintdom  24849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-grpo 20852  df-ablo 20943  df-rngo 21037
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