Theorem iscrngd 13598

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 13597	. 2 |- ( ph -> R e. Ring )
13		eqid 2196	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
14		eqid 2196	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
15	13, 14	mgpbasg 13482	. . . . 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 2229	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
18		eqid 2196	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19	13, 18	mgpplusgg 13480	. . . . 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 2229	. . 3 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
22	17, 21, 5, 6, 9, 10, 11	ismndd 13078	. . 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 13428	. 2 |- ( ph -> (mulGrp ` R ) e. CMnd )
25	13	iscrng 13559	. 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 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   Grpcgrp 13132  CMndccmn 13414  mulGrpcmgp 13476   Ringcrg 13552   CRingccrg 13553

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-cmn 13416  df-mgp 13477  df-ring 13554  df-cring 13555

This theorem is referenced by:  cncrng  14125