Theorem iscrng2 14176

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 2827	. 2 |- ( R e. CRing -> R e. _V )
2		elex 2827	. . 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 2234	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	iscrng 14164	. . 3 |- ( R e. CRing <-> ( R e. Ring /\ (mulGrp ` R ) e. CMnd ) )
6	4	ringmgp 14163	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
7		eqid 2234	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
8		eqid 2234	. . . . . . . 8 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
9	7, 8	iscmn 14027	. . . . . . 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 14087	. . . . . . . . 9 |- ( R e. _V -> B = ( Base ` (mulGrp ` R ) ) )
12		ringcl.t	. . . . . . . . . . . . 13 |- .x. = ( .r ` R )
13	4, 12	mgpplusgg 14085	. . . . . . . . . . . 12 |- ( R e. _V -> .x. = ( +g ` (mulGrp ` R ) ) )
14	13	oveqd 6069	. . . . . . . . . . 11 |- ( R e. _V -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	13	oveqd 6069	. . . . . . . . . . 11 |- ( R e. _V -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
16	14, 15	eqeq12d 2249	. . . . . . . . . 10 |- ( R e. _V -> ( ( x .x. y ) = ( y .x. x ) <-> ( x ( +g ` (mulGrp ` R ) ) y ) = ( y ( +g ` (mulGrp ` R ) ) x ) ) )
17	11, 16	raleqbidv 2759	. . . . . . . . 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 2759	. . . . . . . 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 2205   A.wral 2522   _Vcvv 2815   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308   Mndcmnd 13646  CMndccmn 14018  mulGrpcmgp 14081   Ringcrg 14157   CRingccrg 14158

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-cmn 14020  df-mgp 14082  df-ring 14159  df-cring 14160

This theorem is referenced by:  quscrng  14698