Theorem subrgcrng 14020

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 14019	. . 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 2205	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	1, 4	mgpress 13726	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = (mulGrp ` S ) )
6		eqidd 2206	. . . 4 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) = ( (mulGrp ` R ) ↾s A ) )
7	4	crngmgp 13799	. . . . 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 2205	. . . . . . 7 |- (mulGrp ` S ) = (mulGrp ` S )
10	9	ringmgp 13797	. . . . . 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 2282	. . . 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 13702	. . 3 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> ( (mulGrp ` R ) ↾s A ) e. CMnd )
15	5, 14	eqeltrrd 2283	. 2 |- ( ( R e. CRing /\ A e. (SubRing ` R ) ) -> (mulGrp ` S ) e. CMnd )
16	9	iscrng 13798	. 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 1373   e. wcel 2176   ` cfv 5272  (class class class)co 5946   ↾_s cress 12866   Mndcmnd 13281  CMndccmn 13653  mulGrpcmgp 13715   Ringcrg 13791   CRingccrg 13792  SubRingcsubrg 14012

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873  df-plusg 12955  df-mulr 12956  df-cmn 13655  df-mgp 13716  df-ring 13793  df-cring 13794  df-subrg 14014

This theorem is referenced by:  zringcrng  14387