Theorem subrgcrng 14506

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 14505	. . 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 2238	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	1, 4	mgpress 14205	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = (mulGrp ` S ) )
6		eqidd 2239	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = ( (mulGrp ` R ) ↾s A ) )
7	4	crngmgp 14282	. . . . 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 2238	. . . . . . 7 |- (mulGrp ` S ) = (mulGrp ` S )
10	9	ringmgp 14280	. . . . . 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 2315	. . . 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 14114	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) e. CMnd )
15	5, 14	eqeltrrd 2316	. 2 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> (mulGrp ` S ) e. CMnd )
16	9	iscrng 14281	. 2 |- ( S e. CRing <-> ( S e. Ring /\ (mulGrp ` S ) e. CMnd ) )
17	3, 15, 16	sylanbrc 421	1 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> S e. CRing )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   ↾_s cress 13331   Mndcmnd 13706  CMndccmn 14064  mulGrpcmgp 14194   Ringcrg 14274   CRingccrg 14275  SubRingcsubrg 14498

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-cmn 14066  df-mgp 14195  df-ring 14276  df-cring 14277  df-subrg 14500

This theorem is referenced by:  zringcrng  14899