Theorem subrngintm 14059

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 14051	. . . . 5 |- ( r e. (SubRng ` R ) -> r e. (SubGrp ` R ) )
2	1	ssriv 3201	. . . 4 |- (SubRng ` R ) C_ (SubGrp ` R )
3		sstr 3205	. . . 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 13619	. . 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 3192	. . . . . . 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 3903	. . . . . . . 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 3903	. . . . . . . 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 2206	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
18	17	subrngmcl 14056	. . . . . 6 |- ( ( r e. (SubRng ` R ) /\ x e. r /\ y e. r ) -> ( x ( .r ` R ) y ) e. r )
19	8, 12, 16, 18	syl3anc 1250	. . . . 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 2580	. . . 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 3191	. . . . . . . . 9 |- ( S C_ (SubRng ` R ) -> ( j e. S -> j e. (SubRng ` R ) ) )
22		subrngrcl 14050	. . . . . . . . 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 1843	. . . . . . 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 2776	. . . . . . . 8 |- x e. _V
27	26	a1i 9	. . . . . . 7 |- ( R e. Rng -> x e. _V )
28		mulrslid 13049	. . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
29	28	slotex 12944	. . . . . . 7 |- ( R e. Rng -> ( .r ` R ) e. _V )
30		vex 2776	. . . . . . . 8 |- y e. _V
31	30	a1i 9	. . . . . . 7 |- ( R e. Rng -> y e. _V )
32		ovexg 5996	. . . . . . 7 |- ( ( x e. _V /\ ( .r ` R ) e. _V /\ y e. _V ) -> ( x ( .r ` R ) y ) e. _V )
33	27, 29, 31, 32	syl3anc 1250	. . . . . 6 |- ( R e. Rng -> ( x ( .r ` R ) y ) e. _V )
34		elintg 3902	. . . . . 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 2589	. 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 2206	. . . 4 |- ( Base ` R ) = ( Base ` R )
40	39, 17	issubrng2 14057	. . 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 947	1 |- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1516   e. wcel 2177   A.wral 2485   _Vcvv 2773   C_ wss 3170   |^|cint 3894   ` cfv 5285  (class class class)co 5962   Basecbs 12917   .rcmulr 12995  SubGrpcsubg 13588  Rngcrng 13779  SubRngcsubrng 14044

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-iress 12925  df-plusg 13007  df-mulr 13008  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-grp 13420  df-minusg 13421  df-subg 13591  df-cmn 13707  df-abl 13708  df-mgp 13768  df-rng 13780  df-subrng 14045

This theorem is referenced by:  subrngin  14060