Theorem subrgint 18802

Description: The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)

Assertion

Ref	Expression
subrgint	|- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> |^| S e. (SubRing ` R ) )

Proof of Theorem subrgint

Dummy variables x r y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgsubg 18786	. . . . 5 |- ( r e. (SubRing ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3607	. . . 4 |- (SubRing ` R ) C_ (SubGrp ` R )
3		sstr 3611	. . . 4 |- ( ( S C_ (SubRing ` R ) /\ (SubRing ` R ) C_ (SubGrp ` R ) ) -> S C_ (SubGrp ` R ) )
4	2, 3	mpan2 707	. . 3 |- ( S C_ (SubRing ` R ) -> S C_ (SubGrp ` R ) )
5		subgint 17618	. . 3 |- ( ( S C_ (SubGrp ` R ) /\ S =/= (/) ) -> |^| S e. (SubGrp ` R ) )
6	4, 5	sylan 488	. 2 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> |^| S e. (SubGrp ` R ) )
7		ssel2 3598	. . . . . 6 |- ( ( S C_ (SubRing ` R ) /\ r e. S ) -> r e. (SubRing ` R ) )
8	7	adantlr 751	. . . . 5 |- ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ r e. S ) -> r e. (SubRing ` R ) )
9		eqid 2622	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
10	9	subrg1cl 18788	. . . . 5 |- ( r e. (SubRing ` R ) -> ( 1r ` R ) e. r )
11	8, 10	syl 17	. . . 4 |- ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ r e. S ) -> ( 1r ` R ) e. r )
12	11	ralrimiva 2966	. . 3 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> A. r e. S ( 1r ` R ) e. r )
13		fvex 6201	. . . 4 |- ( 1r ` R ) e. _V
14	13	elint2 4482	. . 3 |- ( ( 1r ` R ) e. |^| S <-> A. r e. S ( 1r ` R ) e. r )
15	12, 14	sylibr 224	. 2 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> ( 1r ` R ) e. |^| S )
16	8	adantlr 751	. . . . . 6 |- ( ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> r e. (SubRing ` R ) )
17		simprl 794	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
18		elinti 4485	. . . . . . . 8 |- ( x e. |^| S -> ( r e. S -> x e. r ) )
19	18	imp 445	. . . . . . 7 |- ( ( x e. |^| S /\ r e. S ) -> x e. r )
20	17, 19	sylan 488	. . . . . 6 |- ( ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> x e. r )
21		simprr 796	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
22		elinti 4485	. . . . . . . 8 |- ( y e. |^| S -> ( r e. S -> y e. r ) )
23	22	imp 445	. . . . . . 7 |- ( ( y e. |^| S /\ r e. S ) -> y e. r )
24	21, 23	sylan 488	. . . . . 6 |- ( ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> y e. r )
25		eqid 2622	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
26	25	subrgmcl 18792	. . . . . 6 |- ( ( r e. (SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
27	16, 20, 24, 26	syl3anc 1326	. . . . 5 |- ( ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> ( x ( .r ` R ) y ) e. r )
28	27	ralrimiva 2966	. . . 4 |- ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> A. r e. S ( x ( .r ` R ) y ) e. r )
29		ovex 6678	. . . . 5 |- ( x ( .r ` R ) y ) e. _V
30	29	elint2 4482	. . . 4 |- ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r )
31	28, 30	sylibr 224	. . 3 |- ( ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( .r ` R ) y ) e. |^| S )
32	31	ralrimivva 2971	. 2 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S )
33		ssn0 3976	. . 3 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> (SubRing ` R ) =/= (/) )
34		n0 3931	. . . 4 |- ( (SubRing ` R ) =/= (/) <-> E. r r e. (SubRing ` R ) )
35		subrgrcl 18785	. . . . 5 |- ( r e. (SubRing ` R ) -> R e. Ring )
36	35	exlimiv 1858	. . . 4 |- ( E. r r e. (SubRing ` R ) -> R e. Ring )
37	34, 36	sylbi 207	. . 3 |- ( (SubRing ` R ) =/= (/) -> R e. Ring )
38		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
39	38, 9, 25	issubrg2 18800	. . 3 |- ( R e. Ring -> ( |^| S e. (SubRing ` R ) <-> ( |^| S e. (SubGrp ` R ) /\ ( 1r ` R ) e. |^| S /\ A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) ) )
40	33, 37, 39	3syl 18	. 2 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> ( |^| S e. (SubRing ` R ) <-> ( |^| S e. (SubGrp ` R ) /\ ( 1r ` R ) e. |^| S /\ A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) ) )
41	6, 15, 32, 40	mpbir3and 1245	1 |- ( ( S C_ (SubRing ` R ) /\ S =/= (/) ) -> |^| S e. (SubRing ` R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   |^|cint 4475   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  SubGrpcsubg 17588   1rcur 18501   Ringcrg 18547  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778

This theorem is referenced by:  subrgin  18803  subrgmre  18804  aspsubrg  19331  rgspncl  37739