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Theorem cnrngo 25081
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 25028 . . 3  |-  +  e.  AbelOp
2 ax-mulf 9568 . . 3  |-  x.  :
( CC  X.  CC )
--> CC
31, 2pm3.2i 455 . 2  |-  (  +  e.  AbelOp  /\  x.  : ( CC  X.  CC ) --> CC )
4 mulass 9576 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
5 adddi 9577 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
6 adddir 9583 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
74, 5, 63jca 1176 . . . 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 2890 . . 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 9546 . . . 4  |-  1  e.  CC
10 mulid2 9590 . . . . . 6  |-  ( y  e.  CC  ->  (
1  x.  y )  =  y )
11 mulid1 9589 . . . . . 6  |-  ( y  e.  CC  ->  (
y  x.  1 )  =  y )
1210, 11jca 532 . . . . 5  |-  ( y  e.  CC  ->  (
( 1  x.  y
)  =  y  /\  ( y  x.  1 )  =  y ) )
1312rgen 2824 . . . 4  |-  A. y  e.  CC  ( ( 1  x.  y )  =  y  /\  ( y  x.  1 )  =  y )
14 oveq1 6289 . . . . . . . 8  |-  ( x  =  1  ->  (
x  x.  y )  =  ( 1  x.  y ) )
1514eqeq1d 2469 . . . . . . 7  |-  ( x  =  1  ->  (
( x  x.  y
)  =  y  <->  ( 1  x.  y )  =  y ) )
16 oveq2 6290 . . . . . . . 8  |-  ( x  =  1  ->  (
y  x.  x )  =  ( y  x.  1 ) )
1716eqeq1d 2469 . . . . . . 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 2903 . . . . 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 3214 . . . 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 11215 . . 3  |-  x.  e.  _V
24 ablogrpo 24962 . . . . . 6  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
251, 24ax-mp 5 . . . . 5  |-  +  e.  GrpOp
26 ax-addf 9567 . . . . . 6  |-  +  :
( CC  X.  CC )
--> CC
2726fdmi 5734 . . . . 5  |-  dom  +  =  ( CC  X.  CC )
2825, 27grporn 24890 . . . 4  |-  CC  =  ran  +
2928isrngo 25056 . . 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 918 1  |-  <.  +  ,  x.  >.  e.  RingOps
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   <.cop 4033    X. cxp 4997   -->wf 5582  (class class class)co 6282   CCcc 9486   1c1 9489    + caddc 9491    x. cmul 9493   GrpOpcgr 24864   AbelOpcablo 24959   RingOpscrngo 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-sub 9803  df-neg 9804  df-grpo 24869  df-ablo 24960  df-rngo 25054
This theorem is referenced by: (None)
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