Theorem subrngintm 13846

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 13838	. . . . 5 ⊢ (𝑟 ∈ (SubRng‘𝑅) → 𝑟 ∈ (SubGrp‘𝑅))
2	1	ssriv 3188	. . . 4 ⊢ (SubRng‘𝑅) ⊆ (SubGrp‘𝑅)
3		sstr 3192	. . . 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 13406	. . 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 3179	. . . . . . 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 3884	. . . . . . . 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 3884	. . . . . . . 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 2196	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
18	17	subrngmcl 13843	. . . . . 6 ⊢ ((𝑟 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑟) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
19	8, 12, 16, 18	syl3anc 1249	. . . . 5 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
20	19	ralrimiva 2570	. . . 4 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
21		ssel 3178	. . . . . . . . 9 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑗 ∈ (SubRng‘𝑅)))
22		subrngrcl 13837	. . . . . . . . 9 ⊢ (𝑗 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
23	21, 22	syl6 33	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
24	23	exlimdv 1833	. . . . . . 7 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (∃𝑗 𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
25	24	imp 124	. . . . . 6 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → 𝑅 ∈ Rng)
26		vex 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
27	26	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑥 ∈ V)
28		mulrslid 12836	. . . . . . . 8 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
29	28	slotex 12732	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (.r‘𝑅) ∈ V)
30		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
31	30	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑦 ∈ V)
32		ovexg 5959	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘𝑅)𝑦) ∈ V)
33	27, 29, 31, 32	syl3anc 1249	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥(.r‘𝑅)𝑦) ∈ V)
34		elintg 3883	. . . . . 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 2579	. 2 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∀𝑥 ∈ ∩ 𝑆∀𝑦 ∈ ∩ 𝑆(𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)
39		eqid 2196	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39, 17	issubrng2 13844	. . 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 946	1 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157  ∩ cint 3875  ‘cfv 5259  (class class class)co 5925  Basecbs 12705  ._rcmulr 12783  SubGrpcsubg 13375  Rngcrng 13566  SubRngcsubrng 13831

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-lttrn 8012  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-subg 13378  df-cmn 13494  df-abl 13495  df-mgp 13555  df-rng 13567  df-subrng 13832

This theorem is referenced by:  subrngin  13847