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Theorem rngoi 21973
Description: The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1  |-  G  =  ( 1st `  R
)
ringi.2  |-  H  =  ( 2nd `  R
)
ringi.3  |-  X  =  ran  G
Assertion
Ref Expression
rngoi  |-  ( R  e.  RingOps  ->  ( ( G  e.  AbelOp  /\  H :
( X  X.  X
) --> X )  /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) ) ) )
Distinct variable groups:    x, y,
z, G    x, H, y, z    x, X, y, z    x, R
Allowed substitution hints:    R( y, z)

Proof of Theorem rngoi
StepHypRef Expression
1 relrngo 21970 . . . . 5  |-  Rel  RingOps
2 1st2nd 6396 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
31, 2mpan 653 . . . 4  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
4 ringi.1 . . . . 5  |-  G  =  ( 1st `  R
)
5 ringi.2 . . . . 5  |-  H  =  ( 2nd `  R
)
64, 5opeq12i 3991 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
73, 6syl6reqr 2489 . . 3  |-  ( R  e.  RingOps  ->  <. G ,  H >.  =  R )
8 id 21 . . 3  |-  ( R  e.  RingOps  ->  R  e.  RingOps )
97, 8eqeltrd 2512 . 2  |-  ( R  e.  RingOps  ->  <. G ,  H >.  e.  RingOps )
10 fvex 5745 . . . 4  |-  ( 2nd `  R )  e.  _V
115, 10eqeltri 2508 . . 3  |-  H  e. 
_V
12 ringi.3 . . . 4  |-  X  =  ran  G
1312isrngo 21971 . . 3  |-  ( H  e.  _V  ->  ( <. G ,  H >.  e.  RingOps  <->  ( ( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X )  /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) ) )
1411, 13ax-mp 5 . 2  |-  ( <. G ,  H >.  e.  RingOps  <->  ( ( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X )  /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
159, 14sylib 190 1  |-  ( R  e.  RingOps  ->  ( ( G  e.  AbelOp  /\  H :
( X  X.  X
) --> X )  /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   <.cop 3819    X. cxp 4879   ran crn 4882   Rel wrel 4886   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   AbelOpcablo 21874   RingOpscrngo 21968
This theorem is referenced by:  rngosm  21974  rngoid  21976  rngoideu  21977  rngodi  21978  rngodir  21979  rngoass  21980  rngoablo  21982  rngorn1eq  22013  rngomndo  22014
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 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-rngo 21969
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