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Theorem iscrngo2 31641
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 31640 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
2 relrngo 25674 . . . . 5  |-  Rel  RingOps
3 1st2nd 6782 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
42, 3mpan 668 . . . 4  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 eleq1 2472 . . . . 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 5169 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
96, 8eqtri 2429 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
109raleqi 3005 . . . . . 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 6245 . . . . . . . . 9  |-  ( x H y )  =  ( x ( 2nd `  R ) y )
1311oveqi 6245 . . . . . . . . 9  |-  ( y H x )  =  ( y ( 2nd `  R ) x )
1412, 13eqeq12i 2420 . . . . . . . 8  |-  ( ( x H y )  =  ( y H x )  <->  ( x
( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) )
159, 14raleqbii 2846 . . . . . . 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 2832 . . . . . 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 5813 . . . . . . 7  |-  ( 1st `  R )  e.  _V
18 fvex 5813 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
19 iscom2 25709 . . . . . . 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 670 . . . . . 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 17 . . 3  |-  ( R  e.  RingOps  ->  ( R  e. 
Com2 
<-> 
A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
2423pm5.32i 635 . 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 367    = wceq 1403    e. wcel 1840   A.wral 2751   _Vcvv 3056   <.cop 3975   ran crn 4941   Rel wrel 4945   ` cfv 5523  (class class class)co 6232   1stc1st 6734   2ndc2nd 6735   RingOpscrngo 25672   Com2ccm2 25707  CRingOpsccring 31638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-1st 6736  df-2nd 6737  df-rngo 25673  df-com2 25708  df-crngo 31639
This theorem is referenced by:  crngocom  31644  crngohomfo  31649
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