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Theorem iscrngo2 29985
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 29984 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
2 relrngo 25041 . . . . 5  |-  Rel  RingOps
3 1st2nd 6820 . . . . 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 2532 . . . . 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 5220 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
96, 8eqtri 2489 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
109raleqi 3055 . . . . . 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 6288 . . . . . . . . 9  |-  ( x H y )  =  ( x ( 2nd `  R ) y )
1311oveqi 6288 . . . . . . . . 9  |-  ( y H x )  =  ( y ( 2nd `  R ) x )
1412, 13eqeq12i 2480 . . . . . . . 8  |-  ( ( x H y )  =  ( y H x )  <->  ( x
( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) )
159, 14raleqbii 2902 . . . . . . 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 2888 . . . . . 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 5867 . . . . . . 7  |-  ( 1st `  R )  e.  _V
18 fvex 5867 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
19 iscom2 25076 . . . . . . 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 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106   <.cop 4026   ran crn 4993   Rel wrel 4997   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   RingOpscrngo 25039   Com2ccm2 25074  CRingOpsccring 29982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-1st 6774  df-2nd 6775  df-rngo 25040  df-com2 25075  df-crngo 29983
This theorem is referenced by:  crngocom  29988  crngohomfo  29993
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