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

Theorem iscrngo2 32151
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 32150 . 2  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
2 relrngo 26097 . . . . 5  |-  Rel  RingOps
3 1st2nd 6851 . . . . 5  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
42, 3mpan 675 . . . 4  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 eleq1 2495 . . . . 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 5078 . . . . . . . 8  |-  ran  G  =  ran  ( 1st `  R
)
96, 8eqtri 2452 . . . . . . 7  |-  X  =  ran  ( 1st `  R
)
109raleqi 3030 . . . . . 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 6316 . . . . . . . . 9  |-  ( x H y )  =  ( x ( 2nd `  R ) y )
1311oveqi 6316 . . . . . . . . 9  |-  ( y H x )  =  ( y ( 2nd `  R ) x )
1412, 13eqeq12i 2443 . . . . . . . 8  |-  ( ( x H y )  =  ( y H x )  <->  ( x
( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) )
159, 14raleqbii 2871 . . . . . . 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 2857 . . . . . 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 5889 . . . . . . 7  |-  ( 1st `  R )  e.  _V
18 fvex 5889 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
19 iscom2 26132 . . . . . . 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 677 . . . . . 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 282 . . . . 5  |-  ( <.
( 1st `  R
) ,  ( 2nd `  R ) >.  e.  Com2  <->  A. x  e.  X  A. y  e.  X  (
x H y )  =  ( y H x ) )
225, 21syl6bb 265 . . . 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 642 . 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 253 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 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776   _Vcvv 3082   <.cop 4003   ran crn 4852   Rel wrel 4856   ` cfv 5599  (class class class)co 6303   1stc1st 6803   2ndc2nd 6804   RingOpscrngo 26095   Com2ccm2 26130  CRingOpsccring 32148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-1st 6805  df-2nd 6806  df-rngo 26096  df-com2 26131  df-crngo 32149
This theorem is referenced by:  crngocom  32154  crngohomfo  32159
  Copyright terms: Public domain W3C validator