Theorem subrngintm 14197

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

Assertion

Ref	Expression
subrngintm	⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (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 14189	. . . . 5 ⊢ (𝑟 ∈ (SubRng‘𝑅) → 𝑟 ∈ (SubGrp‘𝑅))
2	1	ssriv 3228	. . . 4 ⊢ (SubRng‘𝑅) ⊆ (SubGrp‘𝑅)
3		sstr 3232	. . . 4 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ (SubRng‘𝑅) ⊆ (SubGrp‘𝑅)) → 𝑆 ⊆ (SubGrp‘𝑅))
4	2, 3	mpan2 425	. . 3 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → 𝑆 ⊆ (SubGrp‘𝑅))
5		subgintm 13756	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝑅))
6	4, 5	sylan 283	. 2 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝑅))
7		ssel2 3219	. . . . . . 7 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ 𝑟 ∈ 𝑆) → 𝑟 ∈ (SubRng‘𝑅))
8	7	ad4ant14 514	. . . . . 6 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → 𝑟 ∈ (SubRng‘𝑅))
9		simprl 529	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑥 ∈ ∩ 𝑆)
10		elinti 3932	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑆 → (𝑟 ∈ 𝑆 → 𝑥 ∈ 𝑟))
11	10	imp 124	. . . . . . 7 ⊢ ((𝑥 ∈ ∩ 𝑆 ∧ 𝑟 ∈ 𝑆) → 𝑥 ∈ 𝑟)
12	9, 11	sylan 283	. . . . . 6 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → 𝑥 ∈ 𝑟)
13		simprr 531	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑦 ∈ ∩ 𝑆)
14		elinti 3932	. . . . . . . 8 ⊢ (𝑦 ∈ ∩ 𝑆 → (𝑟 ∈ 𝑆 → 𝑦 ∈ 𝑟))
15	14	imp 124	. . . . . . 7 ⊢ ((𝑦 ∈ ∩ 𝑆 ∧ 𝑟 ∈ 𝑆) → 𝑦 ∈ 𝑟)
16	13, 15	sylan 283	. . . . . 6 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → 𝑦 ∈ 𝑟)
17		eqid 2229	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
18	17	subrngmcl 14194	. . . . . 6 ⊢ ((𝑟 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑟) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
19	8, 12, 16, 18	syl3anc 1271	. . . . 5 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
20	19	ralrimiva 2603	. . . 4 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
21		ssel 3218	. . . . . . . . 9 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑗 ∈ (SubRng‘𝑅)))
22		subrngrcl 14188	. . . . . . . . 9 ⊢ (𝑗 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
23	21, 22	syl6 33	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
24	23	exlimdv 1865	. . . . . . 7 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (∃𝑗 𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
25	24	imp 124	. . . . . 6 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → 𝑅 ∈ Rng)
26		vex 2802	. . . . . . . 8 ⊢ 𝑥 ∈ V
27	26	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑥 ∈ V)
28		mulrslid 13186	. . . . . . . 8 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
29	28	slotex 13080	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (.r‘𝑅) ∈ V)
30		vex 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
31	30	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑦 ∈ V)
32		ovexg 6044	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘𝑅)𝑦) ∈ V)
33	27, 29, 31, 32	syl3anc 1271	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥(.r‘𝑅)𝑦) ∈ V)
34		elintg 3931	. . . . . 6 ⊢ ((𝑥(.r‘𝑅)𝑦) ∈ V → ((𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆 ↔ ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟))
35	25, 33, 34	3syl 17	. . . . 5 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ((𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆 ↔ ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟))
36	35	adantr 276	. . . 4 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ((𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆 ↔ ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟))
37	20, 36	mpbird 167	. . 3 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → (𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)
38	37	ralrimivva 2612	. 2 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∀𝑥 ∈ ∩ 𝑆∀𝑦 ∈ ∩ 𝑆(𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)
39		eqid 2229	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39, 17	issubrng2 14195	. . 3 ⊢ (𝑅 ∈ Rng → (∩ 𝑆 ∈ (SubRng‘𝑅) ↔ (∩ 𝑆 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ ∩ 𝑆∀𝑦 ∈ ∩ 𝑆(𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)))
41	25, 40	syl 14	. 2 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → (∩ 𝑆 ∈ (SubRng‘𝑅) ↔ (∩ 𝑆 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ ∩ 𝑆∀𝑦 ∈ ∩ 𝑆(𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)))
42	6, 38, 41	mpbir2and 950	1 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197  ∩ cint 3923  ‘cfv 5321  (class class class)co 6010  Basecbs 13053  ._rcmulr 13132  SubGrpcsubg 13725  Rngcrng 13916  SubRngcsubrng 14182

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 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-pre-ltirr 8127  ax-pre-lttrn 8129  ax-pre-ltadd 8131

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 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-pnf 8199  df-mnf 8200  df-ltxr 8202  df-inn 9127  df-2 9185  df-3 9186  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-0g 13312  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-subg 13728  df-cmn 13844  df-abl 13845  df-mgp 13905  df-rng 13917  df-subrng 14183

This theorem is referenced by:  subrngin  14198