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Theorem rngoi 21925
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 21922 . . . . 5  |-  Rel  RingOps
2 1st2nd 6356 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
31, 2mpan 652 . . . 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 3953 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
73, 6syl6reqr 2459 . . 3  |-  ( R  e.  RingOps  ->  <. G ,  H >.  =  R )
8 id 20 . . 3  |-  ( R  e.  RingOps  ->  R  e.  RingOps )
97, 8eqeltrd 2482 . 2  |-  ( R  e.  RingOps  ->  <. G ,  H >.  e.  RingOps )
10 fvex 5705 . . . 4  |-  ( 2nd `  R )  e.  _V
115, 10eqeltri 2478 . . 3  |-  H  e. 
_V
12 ringi.3 . . . 4  |-  X  =  ran  G
1312isrngo 21923 . . 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 189 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   _Vcvv 2920   <.cop 3781    X. cxp 4839   ran crn 4842   Rel wrel 4846   -->wf 5413   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311   AbelOpcablo 21826   RingOpscrngo 21920
This theorem is referenced by:  rngosm  21926  rngoid  21928  rngoideu  21929  rngodi  21930  rngodir  21931  rngoass  21932  rngoablo  21934  rngorn1eq  21965  rngomndo  21966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-1st 6312  df-2nd 6313  df-rngo 21921
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