Theorem subrgintm 14038

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 14022	. . . . 5 |- ( r e. (SubRing ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3197	. . . 4 |- (SubRing ` R ) C_ (SubGrp ` R )
3		sstr 3201	. . . 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 13567	. . 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 3188	. . . . . 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 2205	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
10	9	subrg1cl 14024	. . . . 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 2579	. . 3 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> A. r e. S ( 1r ` R ) e. r )
13		ssel 3187	. . . . . . 7 |- ( S C_ (SubRing ` R ) -> ( w e. S -> w e. (SubRing ` R ) ) )
14		subrgrcl 14021	. . . . . . 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 1842	. . . . 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 13842	. . . 4 |- ( R e. Ring -> R e. SRing )
19		eqid 2205	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20	19, 9	srgidcl 13771	. . . . 5 |- ( R e. SRing -> ( 1r ` R ) e. ( Base ` R ) )
21		elintg 3893	. . . . 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 3894	. . . . . . . 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 3894	. . . . . . . 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 2205	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
35	34	subrgmcl 14028	. . . . . 6 |- ( ( r e. (SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
36	25, 29, 33, 35	syl3anc 1250	. . . . 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 2579	. . . 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 2266	. . . . . . . 8 |- ( r = w -> ( r e. S <-> w e. S ) )
40	39	cbvexv 1942	. . . . . . 7 |- ( E. r r e. S <-> E. w w e. S )
41	36	elexd 2785	. . . . . . . . 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 1842	. . . . . . 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 3893	. . . . 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 2588	. 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 14036	. . 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 1183	1 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   E.wex 1515   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   |^|cint 3885   ` cfv 5272  (class class class)co 5946   Basecbs 12865   .rcmulr 12943  SubGrpcsubg 13536   1rcur 13754  SRingcsrg 13758   Ringcrg 13791  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-coll 4160  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-reu 2491  df-rmo 2492  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-iun 3929  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-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  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-0g 13123  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-minusg 13369  df-subg 13539  df-cmn 13655  df-abl 13656  df-mgp 13716  df-ur 13755  df-srg 13759  df-ring 13793  df-subrg 14014

This theorem is referenced by:  subrgin  14039