Theorem iscrng2 12991

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 2746	. 2 |- ( R e. CRing -> R e. _V )
2		elex 2746	. . 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 2175	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	iscrng 12979	. . 3 |- ( R e. CRing <-> ( R e. Ring /\ (mulGrp ` R ) e. CMnd ) )
6	4	ringmgp 12978	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
7		eqid 2175	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2175	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9	7, 8	iscmn 12892	. . . . . . 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 12930	. . . . . . . . 9 |- ( R e. _V -> B = ( Base ` (mulGrp ` R ) ) )
12		ringcl.t	. . . . . . . . . . . . 13 |- .x. = ( .r ` R )
13	4, 12	mgpplusgg 12929	. . . . . . . . . . . 12 |- ( R e. _V -> .x. = ( +g ` (mulGrp ` R ) ) )
14	13	oveqd 5882	. . . . . . . . . . 11 |- ( R e. _V -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	13	oveqd 5882	. . . . . . . . . . 11 |- ( R e. _V -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
16	14, 15	eqeq12d 2190	. . . . . . . . . 10 |- ( R e. _V -> ( ( x .x. y ) = ( y .x. x ) <-> ( x ( +g ` (mulGrp ` R ) ) y ) = ( y ( +g ` (mulGrp ` R ) ) x ) ) )
17	11, 16	raleqbidv 2682	. . . . . . . . 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 2682	. . . . . . . 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 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 ) ) ) )
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 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 ) ) )
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 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 ) ) ) )
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 704	1 |- ( R e. CRing <-> ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   A.wral 2453   _Vcvv 2735   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   .rcmulr 12493   Mndcmnd 12682  CMndccmn 12884  mulGrpcmgp 12925   Ringcrg 12972   CRingccrg 12973

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-sets 12435  df-plusg 12505  df-mulr 12506  df-cmn 12886  df-mgp 12926  df-ring 12974  df-cring 12975

This theorem is referenced by: (None)