Theorem iscrng2 14109

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 2815	. 2 |- ( R e. CRing -> R e. _V )
2		elex 2815	. . 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 2231	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	iscrng 14097	. . 3 |- ( R e. CRing <-> ( R e. Ring /\ (mulGrp ` R ) e. CMnd ) )
6	4	ringmgp 14096	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
7		eqid 2231	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2231	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9	7, 8	iscmn 13960	. . . . . . 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 14020	. . . . . . . . 9 |- ( R e. _V -> B = ( Base ` (mulGrp ` R ) ) )
12		ringcl.t	. . . . . . . . . . . . 13 |- .x. = ( .r ` R )
13	4, 12	mgpplusgg 14018	. . . . . . . . . . . 12 |- ( R e. _V -> .x. = ( +g ` (mulGrp ` R ) ) )
14	13	oveqd 6045	. . . . . . . . . . 11 |- ( R e. _V -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	13	oveqd 6045	. . . . . . . . . . 11 |- ( R e. _V -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
16	14, 15	eqeq12d 2246	. . . . . . . . . 10 |- ( R e. _V -> ( ( x .x. y ) = ( y .x. x ) <-> ( x ( +g ` (mulGrp ` R ) ) y ) = ( y ( +g ` (mulGrp ` R ) ) x ) ) )
17	11, 16	raleqbidv 2747	. . . . . . . . 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 2747	. . . . . . . 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 931	. . . . 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 712	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 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   Mndcmnd 13579  CMndccmn 13951  mulGrpcmgp 14014   Ringcrg 14090   CRingccrg 14091

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-cmn 13953  df-mgp 14015  df-ring 14092  df-cring 14093

This theorem is referenced by:  quscrng  14629