Theorem subrgintm 14207

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

Assertion

Ref	Expression
subrgintm	|- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )

Distinct variable groups:   w, R   w, S

Proof of Theorem subrgintm

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

Step	Hyp	Ref	Expression
1		subrgsubg 14191	. . . . 5 |- ( r e. (SubRing ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3228	. . . 4 |- (SubRing ` R ) C_ (SubGrp ` R )
3		sstr 3232	. . . 4 |- ( ( S C_ (SubRing ` R ) /\ (SubRing ` R ) C_ (SubGrp ` R ) ) -> S C_ (SubGrp ` R ) )
4	2, 3	mpan2 425	. . 3 |- ( S C_ (SubRing ` R ) -> S C_ (SubGrp ` R ) )
5		subgintm 13735	. . 3 |- ( ( S C_ (SubGrp ` R ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` R ) )
6	4, 5	sylan 283	. 2 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` R ) )
7		ssel2 3219	. . . . . 6 |- ( ( S C_ (SubRing ` R ) /\ r e. S ) -> r e. (SubRing ` R ) )
8	7	adantlr 477	. . . . 5 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ r e. S ) -> r e. (SubRing ` R ) )
9		eqid 2229	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
10	9	subrg1cl 14193	. . . . 5 |- ( r e. (SubRing ` R ) -> ( 1r ` R ) e. r )
11	8, 10	syl 14	. . . 4 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ r e. S ) -> ( 1r ` R ) e. r )
12	11	ralrimiva 2603	. . 3 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> A. r e. S ( 1r ` R ) e. r )
13		ssel 3218	. . . . . . 7 |- ( S C_ (SubRing ` R ) -> ( w e. S -> w e. (SubRing ` R ) ) )
14		subrgrcl 14190	. . . . . . 7 |- ( w e. (SubRing ` R ) -> R e. Ring )
15	13, 14	syl6 33	. . . . . 6 |- ( S C_ (SubRing ` R ) -> ( w e. S -> R e. Ring ) )
16	15	exlimdv 1865	. . . . 5 |- ( S C_ (SubRing ` R ) -> ( E. w w e. S -> R e. Ring ) )
17	16	imp 124	. . . 4 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> R e. Ring )
18		ringsrg 14010	. . . 4 |- ( R e. Ring -> R e. SRing )
19		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20	19, 9	srgidcl 13939	. . . . 5 |- ( R e. SRing -> ( 1r ` R ) e. ( Base ` R ) )
21		elintg 3931	. . . . 5 |- ( ( 1r ` R ) e. ( Base ` R ) -> ( ( 1r ` R ) e. |^| S <-> A. r e. S ( 1r ` R ) e. r ) )
22	20, 21	syl 14	. . . 4 |- ( R e. SRing -> ( ( 1r ` R ) e. |^| S <-> A. r e. S ( 1r ` R ) e. r ) )
23	17, 18, 22	3syl 17	. . 3 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> ( ( 1r ` R ) e. |^| S <-> A. r e. S ( 1r ` R ) e. r ) )
24	12, 23	mpbird 167	. 2 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> ( 1r ` R ) e. |^| S )
25	8	adantlr 477	. . . . . 6 |- ( ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> r e. (SubRing ` R ) )
26		simprl 529	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
27		elinti 3932	. . . . . . . 8 |- ( x e. |^| S -> ( r e. S -> x e. r ) )
28	27	imp 124	. . . . . . 7 |- ( ( x e. |^| S /\ r e. S ) -> x e. r )
29	26, 28	sylan 283	. . . . . 6 |- ( ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> x e. r )
30		simprr 531	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
31		elinti 3932	. . . . . . . 8 |- ( y e. |^| S -> ( r e. S -> y e. r ) )
32	31	imp 124	. . . . . . 7 |- ( ( y e. |^| S /\ r e. S ) -> y e. r )
33	30, 32	sylan 283	. . . . . 6 |- ( ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> y e. r )
34		eqid 2229	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
35	34	subrgmcl 14197	. . . . . 6 |- ( ( r e. (SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
36	25, 29, 33, 35	syl3anc 1271	. . . . 5 |- ( ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> ( x ( .r ` R ) y ) e. r )
37	36	ralrimiva 2603	. . . 4 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> A. r e. S ( x ( .r ` R ) y ) e. r )
38		simplr 528	. . . . . 6 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> E. w w e. S )
39		eleq1w 2290	. . . . . . . 8 |- ( r = w -> ( r e. S <-> w e. S ) )
40	39	cbvexv 1965	. . . . . . 7 |- ( E. r r e. S <-> E. w w e. S )
41	36	elexd 2813	. . . . . . . . 9 |- ( ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> ( x ( .r ` R ) y ) e. _V )
42	41	ex 115	. . . . . . . 8 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( r e. S -> ( x ( .r ` R ) y ) e. _V ) )
43	42	exlimdv 1865	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( E. r r e. S -> ( x ( .r ` R ) y ) e. _V ) )
44	40, 43	biimtrrid 153	. . . . . 6 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( E. w w e. S -> ( x ( .r ` R ) y ) e. _V ) )
45	38, 44	mpd 13	. . . . 5 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( .r ` R ) y ) e. _V )
46		elintg 3931	. . . . 5 |- ( ( x ( .r ` R ) y ) e. _V -> ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r ) )
47	45, 46	syl 14	. . . 4 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r ) )
48	37, 47	mpbird 167	. . 3 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( .r ` R ) y ) e. |^| S )
49	48	ralrimivva 2612	. 2 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S )
50	19, 9, 34	issubrg2 14205	. . 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 ) ) )
51	17, 50	syl 14	. 2 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. 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 ) ) )
52	6, 24, 49, 51	mpbir3and 1204	1 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   E.wex 1538   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   |^|cint 3923   ` cfv 5318  (class class class)co 6001   Basecbs 13032   .rcmulr 13111  SubGrpcsubg 13704   1rcur 13922  SRingcsrg 13926   Ringcrg 13959  SubRingcsubrg 14181

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-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

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-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-subg 13707  df-cmn 13823  df-abl 13824  df-mgp 13884  df-ur 13923  df-srg 13927  df-ring 13961  df-subrg 14183

This theorem is referenced by:  subrgin  14208