Theorem subrngintm 14229

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 14221	. . . . 5 ⊢ (𝑟 ∈ (SubRng‘𝑅) → 𝑟 ∈ (SubGrp‘𝑅))
2	1	ssriv 3231	. . . 4 ⊢ (SubRng‘𝑅) ⊆ (SubGrp‘𝑅)
3		sstr 3235	. . . 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 13787	. . 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 3222	. . . . . . 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 531	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑥 ∈ ∩ 𝑆)
10		elinti 3937	. . . . . . . 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 533	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑦 ∈ ∩ 𝑆)
14		elinti 3937	. . . . . . . 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 2231	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
18	17	subrngmcl 14226	. . . . . 6 ⊢ ((𝑟 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑟) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
19	8, 12, 16, 18	syl3anc 1273	. . . . 5 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
20	19	ralrimiva 2605	. . . 4 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
21		ssel 3221	. . . . . . . . 9 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑗 ∈ (SubRng‘𝑅)))
22		subrngrcl 14220	. . . . . . . . 9 ⊢ (𝑗 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
23	21, 22	syl6 33	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
24	23	exlimdv 1867	. . . . . . 7 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (∃𝑗 𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
25	24	imp 124	. . . . . 6 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → 𝑅 ∈ Rng)
26		vex 2805	. . . . . . . 8 ⊢ 𝑥 ∈ V
27	26	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑥 ∈ V)
28		mulrslid 13217	. . . . . . . 8 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
29	28	slotex 13111	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (.r‘𝑅) ∈ V)
30		vex 2805	. . . . . . . 8 ⊢ 𝑦 ∈ V
31	30	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑦 ∈ V)
32		ovexg 6052	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘𝑅)𝑦) ∈ V)
33	27, 29, 31, 32	syl3anc 1273	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥(.r‘𝑅)𝑦) ∈ V)
34		elintg 3936	. . . . . 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 2614	. 2 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∀𝑥 ∈ ∩ 𝑆∀𝑦 ∈ ∩ 𝑆(𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)
39		eqid 2231	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39, 17	issubrng2 14227	. . 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 952	1 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ⊆ wss 3200  ∩ cint 3928  ‘cfv 5326  (class class class)co 6018  Basecbs 13084  ._rcmulr 13163  SubGrpcsubg 13756  Rngcrng 13948  SubRngcsubrng 14214

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13087  df-slot 13088  df-base 13090  df-sets 13091  df-iress 13092  df-plusg 13175  df-mulr 13176  df-0g 13343  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-grp 13588  df-minusg 13589  df-subg 13759  df-cmn 13875  df-abl 13876  df-mgp 13937  df-rng 13949  df-subrng 14215

This theorem is referenced by:  subrngin  14230