Theorem iscrng2 14319

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

Step	Hyp	Ref	Expression
1		elex 2833	. 2 |- ( R e. CRing -> R e. _V )
2		elex 2833	. . 3 |- ( R e. Ring -> R e. _V )
3	2	adantr 276	. 2 |- ( ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) -> R e. _V )
4		eqid 2238	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	iscrng 14307	. . 3 |- ( R e. CRing <-> ( R e. Ring /\ (mulGrp ` R ) e. CMnd ) )
6	4	ringmgp 14306	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
7		eqid 2238	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2238	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9	7, 8	iscmn 14096	. . . . . . 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 )
11	4, 10	mgpbasg 14224	. . . . . . . . 9 |- ( R e. _V -> B = ( Base ` (mulGrp ` R ) ) )
12		ringcl.t	. . . . . . . . . . . . 13 |- .x. = ( .r ` R )
13	4, 12	mgpplusgg 14221	. . . . . . . . . . . 12 |- ( R e. _V -> .x. = ( +g ` (mulGrp ` R ) ) )
14	13	oveqd 6102	. . . . . . . . . . 11 |- ( R e. _V -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	13	oveqd 6102	. . . . . . . . . . 11 |- ( R e. _V -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
16	14, 15	eqeq12d 2253	. . . . . . . . . 10 |- ( R e. _V -> ( ( x .x. y ) = ( y .x. x ) <-> ( x ( +g ` (mulGrp ` R ) ) y ) = ( y ( +g ` (mulGrp ` R ) ) x ) ) )
17	11, 16	raleqbidv 2765	. . . . . . . . 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 ) ) )
18	11, 17	raleqbidv 2765	. . . . . . . 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 ) ) )
19	18	anbi2d 468	. . . . . . 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 ) ) ) )
20	9, 19	bitr4id 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 ) ) ) )
21	20	baibd 935	. . . . 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 ) ) )
22	6, 21	sylan2 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 ) ) )
23	22	pm5.32da 456	. . 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 ) ) ) )
24	5, 23	bitrid 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 ) ) ) )
25	1, 3, 24	pm5.21nii 716	1 |- ( R e. CRing <-> ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   .rcmulr 13432   Mndcmnd 13729  CMndccmn 14087  mulGrpcmgp 14217   Ringcrg 14300   CRingccrg 14301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-cmn 14089  df-mgp 14218  df-ring 14302  df-cring 14303

This theorem is used by:  quscrng  14870