Theorem subrgintm 13324

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 13308	. . . . 5 |- ( r e. (SubRing ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3159	. . . 4 |- (SubRing ` R ) C_ (SubGrp ` R )
3		sstr 3163	. . . 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 13011	. . 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 3150	. . . . . 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 2177	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
10	9	subrg1cl 13310	. . . . 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 2550	. . 3 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> A. r e. S ( 1r ` R ) e. r )
13		ssel 3149	. . . . . . 7 |- ( S C_ (SubRing ` R ) -> ( w e. S -> w e. (SubRing ` R ) ) )
14		subrgrcl 13307	. . . . . . 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 1819	. . . . 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 13177	. . . 4 |- ( R e. Ring -> R e. SRing )
19		eqid 2177	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20	19, 9	srgidcl 13112	. . . . 5 |- ( R e. SRing -> ( 1r ` R ) e. ( Base ` R ) )
21		elintg 3852	. . . . 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 3853	. . . . . . . 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 3853	. . . . . . . 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 2177	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
35	34	subrgmcl 13314	. . . . . 6 |- ( ( r e. (SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
36	25, 29, 33, 35	syl3anc 1238	. . . . 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 2550	. . . 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 2238	. . . . . . . 8 |- ( r = w -> ( r e. S <-> w e. S ) )
40	39	cbvexv 1918	. . . . . . 7 |- ( E. r r e. S <-> E. w w e. S )
41	36	elexd 2750	. . . . . . . . 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 1819	. . . . . . 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 3852	. . . . 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 2559	. 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 13322	. . 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 1180	1 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   E.wex 1492   e. wcel 2148   A.wral 2455   _Vcvv 2737   C_ wss 3129   |^|cint 3844   ` cfv 5216  (class class class)co 5874   Basecbs 12456   .rcmulr 12531  SubGrpcsubg 12980   1rcur 13095  SRingcsrg 13099   Ringcrg 13132  SubRingcsubrg 13298

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-subg 12983  df-cmn 13043  df-abl 13044  df-mgp 13084  df-ur 13096  df-srg 13100  df-ring 13134  df-subrg 13300

This theorem is referenced by:  subrgin  13325