Theorem iscrngd 14347

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 14346	. 2 |- ( ph -> R e. Ring )
13		eqid 2238	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
14		eqid 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
15	13, 14	mgpbasg 14224	. . . . 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 2271	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
18		eqid 2238	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19	13, 18	mgpplusgg 14221	. . . . 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 2271	. . 3 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
22	17, 21, 5, 6, 9, 10, 11	ismndd 13750	. . 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 14101	. 2 |- ( ph -> (mulGrp ` R ) e. CMnd )
25	13	iscrng 14307	. 2 |- ( R e. CRing <-> ( R e. Ring /\ (mulGrp ` R ) e. CMnd ) )
26	12, 24, 25	sylanbrc 421	1 |- ( ph -> R e. CRing )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   .rcmulr 13432   Grpcgrp 13805  CMndccmn 14087  mulGrpcmgp 14217   Ringcrg 14300   CRingccrg 14301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-cmn 14089  df-mgp 14218  df-ring 14302  df-cring 14303

This theorem is used by:  cncrng  14906