Theorem subrngintm 14191

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 14183	. . . . 5 |- ( r e. (SubRng ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3228	. . . 4 |- (SubRng ` R ) C_ (SubGrp ` R )
3		sstr 3232	. . . 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 13750	. . 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 3219	. . . . . . 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 529	. . . . . . 7 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
10		elinti 3932	. . . . . . . 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 531	. . . . . . 7 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
14		elinti 3932	. . . . . . . 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 2229	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
18	17	subrngmcl 14188	. . . . . 6 |- ( ( r e. (SubRng ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
19	8, 12, 16, 18	syl3anc 1271	. . . . 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 2603	. . . 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 3218	. . . . . . . . 9 |- ( S C_ (SubRng ` R ) -> ( j e. S -> j e. (SubRng ` R ) ) )
22		subrngrcl 14182	. . . . . . . . 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 1865	. . . . . . 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 2802	. . . . . . . 8 |- x e. _V
27	26	a1i 9	. . . . . . 7 |- ( R e. Rng -> x e. _V )
28		mulrslid 13180	. . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
29	28	slotex 13074	. . . . . . 7 |- ( R e. Rng -> ( .r ` R ) e. _V )
30		vex 2802	. . . . . . . 8 |- y e. _V
31	30	a1i 9	. . . . . . 7 |- ( R e. Rng -> y e. _V )
32		ovexg 6041	. . . . . . 7 |- ( ( x e. _V /\ ( .r ` R ) e. _V /\ y e. _V ) -> ( x ( .r ` R ) y ) e. _V )
33	27, 29, 31, 32	syl3anc 1271	. . . . . 6 |- ( R e. Rng -> ( x ( .r ` R ) y ) e. _V )
34		elintg 3931	. . . . . 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 2612	. 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 2229	. . . 4 |- ( Base ` R ) = ( Base ` R )
40	39, 17	issubrng2 14189	. . 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 950	1 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   |^|cint 3923   ` cfv 5318  (class class class)co 6007   Basecbs 13047   .rcmulr 13126  SubGrpcsubg 13719  Rngcrng 13910  SubRngcsubrng 14176

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-iress 13055  df-plusg 13138  df-mulr 13139  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-subg 13722  df-cmn 13838  df-abl 13839  df-mgp 13899  df-rng 13911  df-subrng 14177

This theorem is referenced by:  subrngin  14192