Theorem subrgcrng 14229

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 14228	. . 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 2229	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	1, 4	mgpress 13934	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = (mulGrp ` S ) )
6		eqidd 2230	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = ( (mulGrp ` R ) ↾s A ) )
7	4	crngmgp 14007	. . . . 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 2229	. . . . . . 7 |- (mulGrp ` S ) = (mulGrp ` S )
10	9	ringmgp 14005	. . . . . 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 2306	. . . 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 13910	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) e. CMnd )
15	5, 14	eqeltrrd 2307	. 2 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> (mulGrp ` S ) e. CMnd )
16	9	iscrng 14006	. 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 1395   e. wcel 2200   ` cfv 5324  (class class class)co 6013   ↾_s cress 13073   Mndcmnd 13489  CMndccmn 13861  mulGrpcmgp 13923   Ringcrg 13999   CRingccrg 14000  SubRingcsubrg 14221

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-cmn 13863  df-mgp 13924  df-ring 14001  df-cring 14002  df-subrg 14223

This theorem is referenced by:  zringcrng  14596