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Theorem iscrng2 13021
Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypotheses
Ref Expression
ringcl.b  |-  B  =  ( Base `  R
)
ringcl.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
iscrng2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\ 
A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x ) ) )
Distinct variable groups:    x, y, B   
x, R, y
Allowed substitution hints:    .x. ( x, y)

Proof of Theorem iscrng2
StepHypRef Expression
1 elex 2748 . 2  |-  ( R  e.  CRing  ->  R  e.  _V )
2 elex 2748 . . 3  |-  ( R  e.  Ring  ->  R  e. 
_V )
32adantr 276 . 2  |-  ( ( R  e.  Ring  /\  A. x  e.  B  A. y  e.  B  (
x  .x.  y )  =  ( y  .x.  x ) )  ->  R  e.  _V )
4 eqid 2177 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
54iscrng 13009 . . 3  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  (mulGrp `  R )  e. CMnd ) )
64ringmgp 13008 . . . . 5  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
7 eqid 2177 . . . . . . . 8  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
8 eqid 2177 . . . . . . . 8  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
97, 8iscmn 12920 . . . . . . 7  |-  ( (mulGrp `  R )  e. CMnd  <->  ( (mulGrp `  R )  e.  Mnd  /\ 
A. x  e.  (
Base `  (mulGrp `  R
) ) A. y  e.  ( Base `  (mulGrp `  R ) ) ( x ( +g  `  (mulGrp `  R ) ) y )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) ) )
10 ringcl.b . . . . . . . . . 10  |-  B  =  ( Base `  R
)
114, 10mgpbasg 12960 . . . . . . . . 9  |-  ( R  e.  _V  ->  B  =  ( Base `  (mulGrp `  R ) ) )
12 ringcl.t . . . . . . . . . . . . 13  |-  .x.  =  ( .r `  R )
134, 12mgpplusgg 12958 . . . . . . . . . . . 12  |-  ( R  e.  _V  ->  .x.  =  ( +g  `  (mulGrp `  R ) ) )
1413oveqd 5886 . . . . . . . . . . 11  |-  ( R  e.  _V  ->  (
x  .x.  y )  =  ( x ( +g  `  (mulGrp `  R ) ) y ) )
1513oveqd 5886 . . . . . . . . . . 11  |-  ( R  e.  _V  ->  (
y  .x.  x )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) )
1614, 15eqeq12d 2192 . . . . . . . . . 10  |-  ( R  e.  _V  ->  (
( x  .x.  y
)  =  ( y 
.x.  x )  <->  ( x
( +g  `  (mulGrp `  R ) ) y )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) ) )
1711, 16raleqbidv 2684 . . . . . . . . 9  |-  ( R  e.  _V  ->  ( A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x )  <->  A. y  e.  ( Base `  (mulGrp `  R ) ) ( x ( +g  `  (mulGrp `  R ) ) y )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) ) )
1811, 17raleqbidv 2684 . . . . . . . 8  |-  ( R  e.  _V  ->  ( A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x )  <->  A. x  e.  ( Base `  (mulGrp `  R ) ) A. y  e.  ( Base `  (mulGrp `  R )
) ( x ( +g  `  (mulGrp `  R ) ) y )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) ) )
1918anbi2d 464 . . . . . . 7  |-  ( R  e.  _V  ->  (
( (mulGrp `  R
)  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  (
x  .x.  y )  =  ( y  .x.  x ) )  <->  ( (mulGrp `  R )  e.  Mnd  /\ 
A. x  e.  (
Base `  (mulGrp `  R
) ) A. y  e.  ( Base `  (mulGrp `  R ) ) ( x ( +g  `  (mulGrp `  R ) ) y )  =  ( y ( +g  `  (mulGrp `  R ) ) x ) ) ) )
209, 19bitr4id 199 . . . . . 6  |-  ( R  e.  _V  ->  (
(mulGrp `  R )  e. CMnd  <-> 
( (mulGrp `  R
)  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  (
x  .x.  y )  =  ( y  .x.  x ) ) ) )
2120baibd 923 . . . . 5  |-  ( ( R  e.  _V  /\  (mulGrp `  R )  e. 
Mnd )  ->  (
(mulGrp `  R )  e. CMnd  <->  A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x ) ) )
226, 21sylan2 286 . . . 4  |-  ( ( R  e.  _V  /\  R  e.  Ring )  -> 
( (mulGrp `  R
)  e. CMnd  <->  A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x ) ) )
2322pm5.32da 452 . . 3  |-  ( R  e.  _V  ->  (
( R  e.  Ring  /\  (mulGrp `  R )  e. CMnd )  <->  ( R  e. 
Ring  /\  A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x ) ) ) )
245, 23bitrid 192 . 2  |-  ( R  e.  _V  ->  ( R  e.  CRing  <->  ( R  e.  Ring  /\  A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x ) ) ) )
251, 3, 24pm5.21nii 704 1  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\ 
A. x  e.  B  A. y  e.  B  ( x  .x.  y )  =  ( y  .x.  x ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737   ` cfv 5212  (class class class)co 5869   Basecbs 12442   +g cplusg 12515   .rcmulr 12516   Mndcmnd 12706  CMndccmn 12912  mulGrpcmgp 12954   Ringcrg 13002   CRingccrg 13003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-pre-ltirr 7911  ax-pre-ltadd 7915
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7981  df-mnf 7982  df-ltxr 7984  df-inn 8906  df-2 8964  df-3 8965  df-ndx 12445  df-slot 12446  df-base 12448  df-sets 12449  df-plusg 12528  df-mulr 12529  df-cmn 12914  df-mgp 12955  df-ring 13004  df-cring 13005
This theorem is referenced by: (None)
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