Theorem iscrngd 14203

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 14202	. 2 |- ( ph -> R e. Ring )
13		eqid 2234	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
14		eqid 2234	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
15	13, 14	mgpbasg 14087	. . . . 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 2267	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
18		eqid 2234	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19	13, 18	mgpplusgg 14085	. . . . 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 2267	. . 3 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
22	17, 21, 5, 6, 9, 10, 11	ismndd 13667	. . 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 14032	. 2 |- ( ph -> (mulGrp ` R ) e. CMnd )
25	13	iscrng 14164	. 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 1005   = wceq 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308   Grpcgrp 13730  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-mgm 13586  df-sgrp 13632  df-mnd 13647  df-cmn 14020  df-mgp 14082  df-ring 14159  df-cring 14160

This theorem is referenced by:  cncrng  14734