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Theorem rngoi 21159
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 21156 . . . . 5  |-  Rel  RingOps
2 1st2nd 6253 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
31, 2mpan 651 . . . 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 3882 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
73, 6syl6reqr 2409 . . 3  |-  ( R  e.  RingOps  ->  <. G ,  H >.  =  R )
8 id 19 . . 3  |-  ( R  e.  RingOps  ->  R  e.  RingOps )
97, 8eqeltrd 2432 . 2  |-  ( R  e.  RingOps  ->  <. G ,  H >.  e.  RingOps )
10 fvex 5622 . . . 4  |-  ( 2nd `  R )  e.  _V
115, 10eqeltri 2428 . . 3  |-  H  e. 
_V
12 ringi.3 . . . 4  |-  X  =  ran  G
1312isrngo 21157 . . 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 8 . 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 188 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   _Vcvv 2864   <.cop 3719    X. cxp 4769   ran crn 4772   Rel wrel 4776   -->wf 5333   ` cfv 5337  (class class class)co 5945   1stc1st 6207   2ndc2nd 6208   AbelOpcablo 21060   RingOpscrngo 21154
This theorem is referenced by:  rngosm  21160  rngoid  21162  rngoideu  21163  rngodi  21164  rngodir  21165  rngoass  21166  rngoablo  21168  rngorn1eq  21199  rngomndo  21200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-1st 6209  df-2nd 6210  df-rngo 21155
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