Theorem subrngintm 14458

Description: The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)

Assertion

Ref	Expression
subrngintm	|- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )

Distinct variable groups:   R, j   S, j

Proof of Theorem subrngintm

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

Step	Hyp	Ref	Expression
1		subrngsubg 14450	. . . . 5 |- ( r e. (SubRng ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3246	. . . 4 |- (SubRng ` R ) C_ (SubGrp ` R )
3		sstr 3250	. . . 4 |- ( ( S C_ (SubRng ` R ) /\ (SubRng ` R ) C_ (SubGrp ` R ) ) -> S C_ (SubGrp ` R ) )
4	2, 3	mpan2 425	. . 3 |- ( S C_ (SubRng ` R ) -> S C_ (SubGrp ` R ) )
5		subgintm 13951	. . 3 |- ( ( S C_ (SubGrp ` R ) /\ E. j j e. S ) -> |^| S e. (SubGrp ` R ) )
6	4, 5	sylan 283	. 2 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubGrp ` R ) )
7		ssel2 3237	. . . . . . 7 |- ( ( S C_ (SubRng ` R ) /\ r e. S ) -> r e. (SubRng ` R ) )
8	7	ad4ant14 514	. . . . . 6 |- ( ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> r e. (SubRng ` R ) )
9		simprl 531	. . . . . . 7 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
10		elinti 3963	. . . . . . . 8 |- ( x e. |^| S -> ( r e. S -> x e. r ) )
11	10	imp 124	. . . . . . 7 |- ( ( x e. |^| S /\ r e. S ) -> x e. r )
12	9, 11	sylan 283	. . . . . 6 |- ( ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> x e. r )
13		simprr 533	. . . . . . 7 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
14		elinti 3963	. . . . . . . 8 |- ( y e. |^| S -> ( r e. S -> y e. r ) )
15	14	imp 124	. . . . . . 7 |- ( ( y e. |^| S /\ r e. S ) -> y e. r )
16	13, 15	sylan 283	. . . . . 6 |- ( ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> y e. r )
17		eqid 2234	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
18	17	subrngmcl 14455	. . . . . 6 |- ( ( r e. (SubRng ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
19	8, 12, 16, 18	syl3anc 1274	. . . . 5 |- ( ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) /\ r e. S ) -> ( x ( .r ` R ) y ) e. r )
20	19	ralrimiva 2617	. . . 4 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> A. r e. S ( x ( .r ` R ) y ) e. r )
21		ssel 3236	. . . . . . . . 9 |- ( S C_ (SubRng ` R ) -> ( j e. S -> j e. (SubRng ` R ) ) )
22		subrngrcl 14449	. . . . . . . . 9 |- ( j e. (SubRng ` R ) -> R e. Rng )
23	21, 22	syl6 33	. . . . . . . 8 |- ( S C_ (SubRng ` R ) -> ( j e. S -> R e. Rng ) )
24	23	exlimdv 1868	. . . . . . 7 |- ( S C_ (SubRng ` R ) -> ( E. j j e. S -> R e. Rng ) )
25	24	imp 124	. . . . . 6 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> R e. Rng )
26		vex 2818	. . . . . . . 8 |- x e. _V
27	26	a1i 9	. . . . . . 7 |- ( R e. Rng -> x e. _V )
28		mulrslid 13429	. . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
29	28	slotex 13323	. . . . . . 7 |- ( R e. Rng -> ( .r ` R ) e. _V )
30		vex 2818	. . . . . . . 8 |- y e. _V
31	30	a1i 9	. . . . . . 7 |- ( R e. Rng -> y e. _V )
32		ovexg 6092	. . . . . . 7 |- ( ( x e. _V /\ ( .r ` R ) e. _V /\ y e. _V ) -> ( x ( .r ` R ) y ) e. _V )
33	27, 29, 31, 32	syl3anc 1274	. . . . . 6 |- ( R e. Rng -> ( x ( .r ` R ) y ) e. _V )
34		elintg 3962	. . . . . 6 |- ( ( x ( .r ` R ) y ) e. _V -> ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r ) )
35	25, 33, 34	3syl 17	. . . . 5 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r ) )
36	35	adantr 276	. . . 4 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( ( x ( .r ` R ) y ) e. |^| S <-> A. r e. S ( x ( .r ` R ) y ) e. r ) )
37	20, 36	mpbird 167	. . 3 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> ( x ( .r ` R ) y ) e. |^| S )
38	37	ralrimivva 2626	. 2 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S )
39		eqid 2234	. . . 4 |- ( Base ` R ) = ( Base ` R )
40	39, 17	issubrng2 14456	. . 3 |- ( R e. Rng -> ( |^| S e. (SubRng ` R ) <-> ( |^| S e. (SubGrp ` R ) /\ A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) ) )
41	25, 40	syl 14	. 2 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> ( |^| S e. (SubRng ` R ) <-> ( |^| S e. (SubGrp ` R ) /\ A. x e. |^| S A. y e. |^| S ( x ( .r ` R ) y ) e. |^| S ) ) )
42	6, 38, 41	mpbir2and 953	1 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1541   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   |^|cint 3954   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375  SubGrpcsubg 13920  Rngcrng 14171  SubRngcsubrng 14443

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923  df-cmn 14039  df-abl 14040  df-mgp 14160  df-rng 14172  df-subrng 14444

This theorem is referenced by:  subrngin  14459