Theorem subrngintm 14629

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 14621	. . . . 5 |- ( r e. (SubRng ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3252	. . . 4 |- (SubRng ` R ) C_ (SubGrp ` R )
3		sstr 3256	. . . 4 |- ( ( S C_ (SubRng ` R ) /\ (SubRng ` R ) C_ (SubGrp ` R ) ) -> S C_ (SubGrp ` R ) )
4	2, 3	mpan2 429	. . 3 |- ( S C_ (SubRng ` R ) -> S C_ (SubGrp ` R ) )
5		subgintm 14060	. . 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 3243	. . . . . . 7 |- ( ( S C_ (SubRng ` R ) /\ r e. S ) -> r e. (SubRng ` R ) )
8	7	ad4ant14 518	. . . . . 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 535	. . . . . . 7 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> x e. |^| S )
10		elinti 3979	. . . . . . . 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 537	. . . . . . 7 |- ( ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) /\ ( x e. |^| S /\ y e. |^| S ) ) -> y e. |^| S )
14		elinti 3979	. . . . . . . 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 2238	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
18	17	subrngmcl 14626	. . . . . 6 |- ( ( r e. (SubRng ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
19	8, 12, 16, 18	syl3anc 1278	. . . . 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 2623	. . . 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 3242	. . . . . . . . 9 |- ( S C_ (SubRng ` R ) -> ( j e. S -> j e. (SubRng ` R ) ) )
22		subrngrcl 14620	. . . . . . . . 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 1872	. . . . . . 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 2824	. . . . . . . 8 |- x e. _V
27	26	a1i 9	. . . . . . 7 |- ( R e. Rng -> x e. _V )
28		mulrslid 13545	. . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
29	28	slotex 13437	. . . . . . 7 |- ( R e. Rng -> ( .r ` R ) e. _V )
30		vex 2824	. . . . . . . 8 |- y e. _V
31	30	a1i 9	. . . . . . 7 |- ( R e. Rng -> y e. _V )
32		ovexg 6122	. . . . . . 7 |- ( ( x e. _V /\ ( .r ` R ) e. _V /\ y e. _V ) -> ( x ( .r ` R ) y ) e. _V )
33	27, 29, 31, 32	syl3anc 1278	. . . . . 6 |- ( R e. Rng -> ( x ( .r ` R ) y ) e. _V )
34		elintg 3978	. . . . . 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 2632	. 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 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
40	39, 17	issubrng2 14627	. . 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 957	1 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   |^|cint 3970   ` cfv 5380  (class class class)co 6088   Basecbs 13410   .rcmulr 13491  SubGrpcsubg 14029  Rngcrng 14338  SubRngcsubrng 14614

This proof depends on 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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-subg 14032  df-cmn 14179  df-abl 14180  df-mgp 14325  df-rng 14339  df-subrng 14615

This theorem is used by:  subrngin  14630