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Theorem iscrngo2 28798
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 28797 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
2 relrngo 23864 . . . . 5  |-  Rel  RingOps
3 1st2nd 6620 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
42, 3mpan 670 . . . 4  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 eleq1 2503 . . . . 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 5066 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
96, 8eqtri 2463 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
109raleqi 2921 . . . . . 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 . . . . . . . . . 10  |-  H  =  ( 2nd `  R
)
1211oveqi 6104 . . . . . . . . 9  |-  ( x H y )  =  ( x ( 2nd `  R ) y )
1311oveqi 6104 . . . . . . . . 9  |-  ( y H x )  =  ( y ( 2nd `  R ) x )
1412, 13eqeq12i 2456 . . . . . . . 8  |-  ( ( x H y )  =  ( y H x )  <->  ( x
( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) )
159, 14raleqbii 2745 . . . . . . 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 ) )
1615ralbii 2739 . . . . . 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 ) )
17 fvex 5701 . . . . . . 7  |-  ( 1st `  R )  e.  _V
18 fvex 5701 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
19 iscom2 23899 . . . . . . 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 ) ) )
2017, 18, 19mp2an 672 . . . . . 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 ) )
2110, 16, 203bitr4ri 278 . . . . 5  |-  ( <.
( 1st `  R
) ,  ( 2nd `  R ) >.  e.  Com2  <->  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) )
225, 21syl6bb 261 . . . 4  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  Com2  <->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
234, 22syl 16 . . 3  |-  ( R  e.  RingOps  ->  ( R  e. 
Com2 
<-> 
A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
2423pm5.32i 637 . 2  |-  ( ( R  e.  RingOps  /\  R  e.  Com2 )  <->  ( R  e.  RingOps  /\  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
251, 24bitri 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   <.cop 3883   ran crn 4841   Rel wrel 4845   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   RingOpscrngo 23862   Com2ccm2 23897  CRingOpsccring 28795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-1st 6577  df-2nd 6578  df-rngo 23863  df-com2 23898  df-crngo 28796
This theorem is referenced by:  crngocom  28801  crngohomfo  28806
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