Theorem subrngintm 13916

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 13908	. . . . 5 |- ( r e. (SubRng ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3196	. . . 4 |- (SubRng ` R ) C_ (SubGrp ` R )
3		sstr 3200	. . . 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 13476	. . 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 3187	. . . . . . 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 3893	. . . . . . . 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 3893	. . . . . . . 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 2204	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
18	17	subrngmcl 13913	. . . . . 6 |- ( ( r e. (SubRng ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
19	8, 12, 16, 18	syl3anc 1249	. . . . 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 2578	. . . 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 3186	. . . . . . . . 9 |- ( S C_ (SubRng ` R ) -> ( j e. S -> j e. (SubRng ` R ) ) )
22		subrngrcl 13907	. . . . . . . . 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 1841	. . . . . . 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 2774	. . . . . . . 8 |- x e. _V
27	26	a1i 9	. . . . . . 7 |- ( R e. Rng -> x e. _V )
28		mulrslid 12906	. . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
29	28	slotex 12801	. . . . . . 7 |- ( R e. Rng -> ( .r ` R ) e. _V )
30		vex 2774	. . . . . . . 8 |- y e. _V
31	30	a1i 9	. . . . . . 7 |- ( R e. Rng -> y e. _V )
32		ovexg 5977	. . . . . . 7 |- ( ( x e. _V /\ ( .r ` R ) e. _V /\ y e. _V ) -> ( x ( .r ` R ) y ) e. _V )
33	27, 29, 31, 32	syl3anc 1249	. . . . . 6 |- ( R e. Rng -> ( x ( .r ` R ) y ) e. _V )
34		elintg 3892	. . . . . 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 2587	. 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 2204	. . . 4 |- ( Base ` R ) = ( Base ` R )
40	39, 17	issubrng2 13914	. . 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 946	1 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1514   e. wcel 2175   A.wral 2483   _Vcvv 2771   C_ wss 3165   |^|cint 3884   ` cfv 5270  (class class class)co 5943   Basecbs 12774   .rcmulr 12852  SubGrpcsubg 13445  Rngcrng 13636  SubRngcsubrng 13901

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-iress 12782  df-plusg 12864  df-mulr 12865  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278  df-subg 13448  df-cmn 13564  df-abl 13565  df-mgp 13625  df-rng 13637  df-subrng 13902

This theorem is referenced by:  subrngin  13917