Theorem subrgintm 14524

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 14508	. . . . 5 |- ( r e. (SubRing ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3252	. . . 4 |- (SubRing ` R ) C_ (SubGrp ` R )
3		sstr 3256	. . . 4 |- ( ( S C_ (SubRing ` R ) /\ (SubRing ` R ) C_ (SubGrp ` R ) ) -> S C_ (SubGrp ` R ) )
4	2, 3	mpan2 429	. . 3 |- ( S C_ (SubRing ` R ) -> S C_ (SubGrp ` R ) )
5		subgintm 13978	. . 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 3243	. . . . . 6 |- ( ( S C_ (SubRing ` R ) /\ r e. S ) -> r e. (SubRing ` R ) )
8	7	adantlr 481	. . . . 5 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ r e. S ) -> r e. (SubRing ` R ) )
9		eqid 2238	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
10	9	subrg1cl 14510	. . . . 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 2623	. . 3 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> A. r e. S ( 1r ` R ) e. r )
13		ssel 3242	. . . . . . 7 |- ( S C_ (SubRing ` R ) -> ( w e. S -> w e. (SubRing ` R ) ) )
14		subrgrcl 14507	. . . . . . 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 1872	. . . . 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 14325	. . . 4 |- ( R e. Ring -> R e. SRing )
19		eqid 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
20	19, 9	srgidcl 14254	. . . . 5 |- ( R e. SRing -> ( 1r ` R ) e. ( Base ` R ) )
21		elintg 3973	. . . . 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 481	. . . . . 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 535	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
27		elinti 3974	. . . . . . . 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 537	. . . . . . 7 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
31		elinti 3974	. . . . . . . 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 2238	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
35	34	subrgmcl 14514	. . . . . 6 |- ( ( r e. (SubRing ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
36	25, 29, 33, 35	syl3anc 1278	. . . . 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 2623	. . . 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 533	. . . . . 6 |- ( ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> E. w w e. S )
39		eleq1w 2299	. . . . . . . 8 |- ( r = w -> ( r e. S <-> w e. S ) )
40	39	cbvexv 1974	. . . . . . 7 |- ( E. r r e. S <-> E. w w e. S )
41	36	elexd 2835	. . . . . . . . 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 1872	. . . . . . 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 3973	. . . . 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 2632	. 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 14522	. . 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 1211	1 |- ( ( S C_ (SubRing ` R ) /\ E. w w e. S ) -> |^| S e. (SubRing ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   E.wex 1545   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   |^|cint 3965   ` cfv 5372  (class class class)co 6075   Basecbs 13330   .rcmulr 13409  SubGrpcsubg 13947   1rcur 14237  SRingcsrg 14241   Ringcrg 14274  SubRingcsubrg 14498

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-ur 14238  df-srg 14242  df-ring 14276  df-subrg 14500

This theorem is referenced by:  subrgin  14525