Theorem subrgcrng 14320

Description: A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypothesis

Ref	Expression
subrgring.1	|- S = ( R ↾s A )

Assertion

Ref	Expression
subrgcrng	|- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> S e. CRing )

Proof of Theorem subrgcrng

Step	Hyp	Ref	Expression
1		subrgring.1	. . . 4 |- S = ( R ↾s A )
2	1	subrgring 14319	. . 3 |- ( A e. (SubRing ` R ) -> S e. Ring )
3	2	adantl 277	. 2 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> S e. Ring )
4		eqid 2231	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	1, 4	mgpress 14025	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = (mulGrp ` S ) )
6		eqidd 2232	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = ( (mulGrp ` R ) ↾s A ) )
7	4	crngmgp 14098	. . . . 5 |- ( R e. CRing -> (mulGrp ` R ) e. CMnd )
8	7	adantr 276	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> (mulGrp ` R ) e. CMnd )
9		eqid 2231	. . . . . . 7 |- (mulGrp ` S ) = (mulGrp ` S )
10	9	ringmgp 14096	. . . . . 6 |- ( S e. Ring -> (mulGrp ` S ) e. Mnd )
11	3, 10	syl 14	. . . . 5 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> (mulGrp ` S ) e. Mnd )
12	5, 11	eqeltrd 2308	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) e. Mnd )
13		simpr 110	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> A e. (SubRing ` R ) )
14	6, 8, 12, 13	subcmnd 14000	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) e. CMnd )
15	5, 14	eqeltrrd 2309	. 2 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> (mulGrp ` S ) e. CMnd )
16	9	iscrng 14097	. 2 |- ( S e. CRing <-> ( S e. Ring /\ (mulGrp ` S ) e. CMnd ) )
17	3, 15, 16	sylanbrc 417	1 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> S e. CRing )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   ↾_s cress 13163   Mndcmnd 13579  CMndccmn 13951  mulGrpcmgp 14014   Ringcrg 14090   CRingccrg 14091  SubRingcsubrg 14312

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-cmn 13953  df-mgp 14015  df-ring 14092  df-cring 14093  df-subrg 14314

This theorem is referenced by:  zringcrng  14688