Theorem subrngintm 14503

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 14495	. . . . 5 ⊢ (𝑟 ∈ (SubRng‘𝑅) → 𝑟 ∈ (SubGrp‘𝑅))
2	1	ssriv 3252	. . . 4 ⊢ (SubRng‘𝑅) ⊆ (SubGrp‘𝑅)
3		sstr 3256	. . . 4 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ (SubRng‘𝑅) ⊆ (SubGrp‘𝑅)) → 𝑆 ⊆ (SubGrp‘𝑅))
4	2, 3	mpan2 429	. . 3 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → 𝑆 ⊆ (SubGrp‘𝑅))
5		subgintm 13984	. . 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 3243	. . . . . . 7 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ 𝑟 ∈ 𝑆) → 𝑟 ∈ (SubRng‘𝑅))
8	7	ad4ant14 518	. . . . . 6 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → 𝑟 ∈ (SubRng‘𝑅))
9		simprl 535	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑥 ∈ ∩ 𝑆)
10		elinti 3977	. . . . . . . 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 537	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑦 ∈ ∩ 𝑆)
14		elinti 3977	. . . . . . . 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 2238	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
18	17	subrngmcl 14500	. . . . . 6 ⊢ ((𝑟 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝑟 ∧ 𝑦 ∈ 𝑟) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
19	8, 12, 16, 18	syl3anc 1278	. . . . 5 ⊢ ((((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑟 ∈ 𝑆) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
20	19	ralrimiva 2623	. . . 4 ⊢ (((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑟 ∈ 𝑆 (𝑥(.r‘𝑅)𝑦) ∈ 𝑟)
21		ssel 3242	. . . . . . . . 9 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑗 ∈ (SubRng‘𝑅)))
22		subrngrcl 14494	. . . . . . . . 9 ⊢ (𝑗 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
23	21, 22	syl6 33	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
24	23	exlimdv 1872	. . . . . . 7 ⊢ (𝑆 ⊆ (SubRng‘𝑅) → (∃𝑗 𝑗 ∈ 𝑆 → 𝑅 ∈ Rng))
25	24	imp 124	. . . . . 6 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → 𝑅 ∈ Rng)
26		vex 2824	. . . . . . . 8 ⊢ 𝑥 ∈ V
27	26	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑥 ∈ V)
28		mulrslid 13469	. . . . . . . 8 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
29	28	slotex 13362	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (.r‘𝑅) ∈ V)
30		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
31	30	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑦 ∈ V)
32		ovexg 6113	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘𝑅)𝑦) ∈ V)
33	27, 29, 31, 32	syl3anc 1278	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥(.r‘𝑅)𝑦) ∈ V)
34		elintg 3976	. . . . . 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 2632	. 2 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∀𝑥 ∈ ∩ 𝑆∀𝑦 ∈ ∩ 𝑆(𝑥(.r‘𝑅)𝑦) ∈ ∩ 𝑆)
39		eqid 2238	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39, 17	issubrng2 14501	. . 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 957	1 ⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  ∩ cint 3968  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  ._rcmulr 13415  SubGrpcsubg 13953  Rngcrng 14214  SubRngcsubrng 14488

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-subg 13956  df-cmn 14072  df-abl 14073  df-mgp 14201  df-rng 14215  df-subrng 14489

This theorem is referenced by:  subrngin  14504