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Theorem cnrngo 25277
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 25224 . . 3  |-  +  e.  AbelOp
2 ax-mulf 9575 . . 3  |-  x.  :
( CC  X.  CC )
--> CC
31, 2pm3.2i 455 . 2  |-  (  +  e.  AbelOp  /\  x.  : ( CC  X.  CC ) --> CC )
4 mulass 9583 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
5 adddi 9584 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
6 adddir 9590 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
74, 5, 63jca 1177 . . . 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 2869 . . 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 9553 . . . 4  |-  1  e.  CC
10 mulid2 9597 . . . . . 6  |-  ( y  e.  CC  ->  (
1  x.  y )  =  y )
11 mulid1 9596 . . . . . 6  |-  ( y  e.  CC  ->  (
y  x.  1 )  =  y )
1210, 11jca 532 . . . . 5  |-  ( y  e.  CC  ->  (
( 1  x.  y
)  =  y  /\  ( y  x.  1 )  =  y ) )
1312rgen 2803 . . . 4  |-  A. y  e.  CC  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y )
14 oveq1 6288 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
1514eqeq1d 2445 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  =  y  <->  ( 1  x.  y )  =  y ) )
16 oveq2 6289 . . . . . . . 8  |-  ( x  =  1  ->  (
y  x.  x )  =  ( y  x.  1 ) )
1716eqeq1d 2445 . . . . . . 7  |-  ( x  =  1  ->  (
( y  x.  x
)  =  y  <->  ( y  x.  1 )  =  y ) )
1815, 17anbi12d 710 . . . . . 6  |-  ( x  =  1  ->  (
( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )  <->  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y ) ) )
1918ralbidv 2882 . . . . 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 3196 . . . 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 672 . . 3  |-  E. x  e.  CC  A. y  e.  CC  ( ( x  x.  y )  =  y  /\  ( y  x.  x )  =  y )
228, 21pm3.2i 455 . 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 11228 . . 3  |-  x.  e.  _V
24 ablogrpo 25158 . . . . . 6  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
251, 24ax-mp 5 . . . . 5  |-  +  e.  GrpOp
26 ax-addf 9574 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
2726fdmi 5726 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
2825, 27grporn 25086 . . . 4  |-  CC  =  ran  +
2928isrngo 25252 . . 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 920 1  |-  <.  +  ,  x.  >.  e.  RingOps
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095   <.cop 4020    X. cxp 4987   -->wf 5574  (class class class)co 6281   CCcc 9493   1c1 9496    + caddc 9498    x. cmul 9500   GrpOpcgr 25060   AbelOpcablo 25155   RingOpscrngo 25249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-sub 9812  df-neg 9813  df-grpo 25065  df-ablo 25156  df-rngo 25250
This theorem is referenced by: (None)
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