Theorem subrgintm 14338

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 14322	. . . . 5 |- ( r e. (SubRing ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3232	. . . 4 |- (SubRing ` R ) C_ (SubGrp ` R )
3		sstr 3236	. . . 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 13865	. . 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 3223	. . . . . 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 2231	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
10	9	subrg1cl 14324	. . . . 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 2606	. . 3 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> A. r e. S ( 1r ` R ) e. r )
13		ssel 3222	. . . . . . 7 |- ( S C_ (SubRing ` R ) -> ( w e. S -> w e. (SubRing ` R ) ) )
14		subrgrcl 14321	. . . . . . 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 1867	. . . . 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 14141	. . . 4 |- ( R e. Ring -> R e. SRing )
19		eqid 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20	19, 9	srgidcl 14070	. . . . 5 |- ( R e. SRing -> ( 1r ` R ) e. ( Base ` R ) )
21		elintg 3941	. . . . 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 531	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
27		elinti 3942	. . . . . . . 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 533	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
31		elinti 3942	. . . . . . . 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 2231	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
35	34	subrgmcl 14328	. . . . . 6 |- ( ( r e. (SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
36	25, 29, 33, 35	syl3anc 1274	. . . . 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 2606	. . . 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 529	. . . . . 6 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> E. w w e. S )
39		eleq1w 2292	. . . . . . . 8 |- ( r = w -> ( r e. S <-> w e. S ) )
40	39	cbvexv 1967	. . . . . . 7 |- ( E. r r e. S <-> E. w w e. S )
41	36	elexd 2817	. . . . . . . . 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 1867	. . . . . . 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 3941	. . . . 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 2615	. 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 14336	. . 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 1207	1 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   E.wex 1541   e. wcel 2202   A.wral 2511   _Vcvv 2803   C_ wss 3201   |^|cint 3933   ` cfv 5333  (class class class)co 6028   Basecbs 13162   .rcmulr 13241  SubGrpcsubg 13834   1rcur 14053  SRingcsrg 14057   Ringcrg 14090  SubRingcsubrg 14312

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-subg 13837  df-cmn 13953  df-abl 13954  df-mgp 14015  df-ur 14054  df-srg 14058  df-ring 14092  df-subrg 14314

This theorem is referenced by:  subrgin  14339