Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscrngo2 Unicode version

Theorem iscrngo2 26623
Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
iscring2.1  |-  G  =  ( 1st `  R
)
iscring2.2  |-  H  =  ( 2nd `  R
)
iscring2.3  |-  X  =  ran  G
Assertion
Ref Expression
iscrngo2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) ) )
Distinct variable groups:    x, R, y    x, X, y
Allowed substitution hints:    G( x, y)    H( x, y)

Proof of Theorem iscrngo2
StepHypRef Expression
1 iscrngo 26622 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
2 relrngo 21044 . . . . 5  |-  Rel  RingOps
3 1st2nd 6166 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
42, 3mpan 651 . . . 4  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 eleq1 2343 . . . . 5  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  Com2  <->  <. ( 1st `  R ) ,  ( 2nd `  R )
>.  e.  Com2 ) )
6 iscring2.3 . . . . . . . 8  |-  X  =  ran  G
7 iscring2.1 . . . . . . . . 9  |-  G  =  ( 1st `  R
)
87rneqi 4905 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
96, 8eqtri 2303 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
109raleqi 2740 . . . . . 6  |-  ( A. x  e.  X  A. y  e.  ran  ( 1st `  R ) ( x ( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x )  <->  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R ) ( x ( 2nd `  R ) y )  =  ( y ( 2nd `  R ) x ) )
11 iscring2.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
1211oveqi 5871 . . . . . . . . . 10  |-  ( x H y )  =  ( x ( 2nd `  R ) y )
1311oveqi 5871 . . . . . . . . . 10  |-  ( y H x )  =  ( y ( 2nd `  R ) x )
1412, 13eqeq12i 2296 . . . . . . . . 9  |-  ( ( x H y )  =  ( y H x )  <->  ( x
( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) )
1514ralbii 2567 . . . . . . . 8  |-  ( A. y  e.  X  (
x H y )  =  ( y H x )  <->  A. y  e.  X  ( x
( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) )
169raleqi 2740 . . . . . . . 8  |-  ( A. y  e.  X  (
x ( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x )  <->  A. y  e.  ran  ( 1st `  R ) ( x ( 2nd `  R ) y )  =  ( y ( 2nd `  R ) x ) )
1715, 16bitri 240 . . . . . . 7  |-  ( A. y  e.  X  (
x H y )  =  ( y H x )  <->  A. y  e.  ran  ( 1st `  R
) ( x ( 2nd `  R ) y )  =  ( y ( 2nd `  R
) x ) )
1817ralbii 2567 . . . . . 6  |-  ( A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x )  <->  A. x  e.  X  A. y  e.  ran  ( 1st `  R
) ( x ( 2nd `  R ) y )  =  ( y ( 2nd `  R
) x ) )
19 fvex 5539 . . . . . . 7  |-  ( 1st `  R )  e.  _V
20 fvex 5539 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
21 iscom2 21079 . . . . . . 7  |-  ( ( ( 1st `  R
)  e.  _V  /\  ( 2nd `  R )  e.  _V )  -> 
( <. ( 1st `  R
) ,  ( 2nd `  R ) >.  e.  Com2  <->  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R
) ( x ( 2nd `  R ) y )  =  ( y ( 2nd `  R
) x ) ) )
2219, 20, 21mp2an 653 . . . . . 6  |-  ( <.
( 1st `  R
) ,  ( 2nd `  R ) >.  e.  Com2  <->  A. x  e.  ran  ( 1st `  R ) A. y  e.  ran  ( 1st `  R
) ( x ( 2nd `  R ) y )  =  ( y ( 2nd `  R
) x ) )
2310, 18, 223bitr4ri 269 . . . . 5  |-  ( <.
( 1st `  R
) ,  ( 2nd `  R ) >.  e.  Com2  <->  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) )
245, 23syl6bb 252 . . . 4  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  Com2  <->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
254, 24syl 15 . . 3  |-  ( R  e.  RingOps  ->  ( R  e. 
Com2 
<-> 
A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
2625pm5.32i 618 . 2  |-  ( ( R  e.  RingOps  /\  R  e.  Com2 )  <->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
271, 26bitri 240 1  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   ran crn 4690   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RingOpscrngo 21042   Com2ccm2 21077  CRingOpsccring 26620
This theorem is referenced by:  crngocom  26626  crngohomfo  26631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043  df-com2 21078  df-crngo 26621
  Copyright terms: Public domain W3C validator