Theorem iscrngd 13013

Description: Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
isringd.b	|- ( ph -> B = ( Base ` R ) )
isringd.p	|- ( ph -> .+ = ( +g ` R ) )
isringd.t	|- ( ph -> .x. = ( .r ` R ) )
isringd.g	|- ( ph -> R e. Grp )
isringd.c	|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B )
isringd.a	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )
isringd.d	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) )
isringd.e	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
isringd.u	|- ( ph -> .1. e. B )
isringd.i	|- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x )
isringd.h	|- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x )
iscrngd.c	|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) = ( y .x. x ) )

Assertion

Ref	Expression
iscrngd	|- ( ph -> R e. CRing )

Distinct variable groups:   x, .1.   x, y, z, B   ph, x, y, z   x, R, y, z

Allowed substitution hints:   .+ ( x, y, z)   .x. ( x, y, z)   .1. ( y, z)

Proof of Theorem iscrngd

Step	Hyp	Ref	Expression
1		isringd.b	. . 3 |- ( ph -> B = ( Base ` R ) )
2		isringd.p	. . 3 |- ( ph -> .+ = ( +g ` R ) )
3		isringd.t	. . 3 |- ( ph -> .x. = ( .r ` R ) )
4		isringd.g	. . 3 |- ( ph -> R e. Grp )
5		isringd.c	. . 3 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B )
6		isringd.a	. . 3 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )
7		isringd.d	. . 3 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) )
8		isringd.e	. . 3 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
9		isringd.u	. . 3 |- ( ph -> .1. e. B )
10		isringd.i	. . 3 |- ( ( ph /\ x e. B ) -> ( .1. .x. x ) = x )
11		isringd.h	. . 3 |- ( ( ph /\ x e. B ) -> ( x .x. .1. ) = x )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	isringd 13012	. 2 |- ( ph -> R e. Ring )
13		eqid 2175	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
14		eqid 2175	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
15	13, 14	mgpbasg 12930	. . . . 5 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
16	12, 15	syl 14	. . . 4 |- ( ph -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
17	1, 16	eqtrd 2208	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
18		eqid 2175	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19	13, 18	mgpplusgg 12929	. . . . 5 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
20	12, 19	syl 14	. . . 4 |- ( ph -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
21	3, 20	eqtrd 2208	. . 3 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
22	17, 21, 5, 6, 9, 10, 11	ismndd 12703	. . 3 |- ( ph -> (mulGrp ` R ) e. Mnd )
23		iscrngd.c	. . 3 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) = ( y .x. x ) )
24	17, 21, 22, 23	iscmnd 12897	. 2 |- ( ph -> (mulGrp ` R ) e. CMnd )
25	13	iscrng 12979	. 2 |- ( R e. CRing <-> ( R e. Ring /\ (mulGrp ` R ) e. CMnd ) )
26	12, 24, 25	sylanbrc 417	1 |- ( ph -> R e. CRing )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2146   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   .rcmulr 12493   Grpcgrp 12738  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-mgm 12640  df-sgrp 12673  df-mnd 12683  df-cmn 12886  df-mgp 12926  df-ring 12974  df-cring 12975

This theorem is referenced by: (None)