Theorem subdrgint 20826

Description: The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)

Hypotheses

Ref	Expression
subdrgint.1	⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)
subdrgint.2	⊢ (𝜑 → 𝑅 ∈ DivRing)
subdrgint.3	⊢ (𝜑 → 𝑆 ⊆ (SubRing‘𝑅))
subdrgint.4	⊢ (𝜑 → 𝑆 ≠ ∅)
subdrgint.5	⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑅 ↾s 𝑠) ∈ DivRing)

Assertion

Ref	Expression
subdrgint	⊢ (𝜑 → 𝐿 ∈ DivRing)

Distinct variable groups:   𝐿,𝑠   𝑅,𝑠   𝑆,𝑠   𝜑,𝑠

Proof of Theorem subdrgint

Step	Hyp	Ref	Expression
1		subdrgint.3	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (SubRing‘𝑅))
2		subdrgint.4	. . . 4 ⊢ (𝜑 → 𝑆 ≠ ∅)
3		subrgint 20623	. . . 4 ⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅))
4	1, 2, 3	syl2anc 583	. . 3 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubRing‘𝑅))
5		subdrgint.1	. . . 4 ⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)
6	5	subrgring 20602	. . 3 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐿 ∈ Ring)
8	5	fveq2i 6923	. . . 4 ⊢ (mulGrp‘𝐿) = (mulGrp‘(𝑅 ↾s ∩ 𝑆))
9	8	oveq1i 7458	. . 3 ⊢ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)}))
10		subdrgint.2	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ DivRing)
11		eqid 2740	. . . . . . . 8 ⊢ (𝑅 ↾s ∩ 𝑆) = (𝑅 ↾s ∩ 𝑆)
12		eqid 2740	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
13	11, 12	mgpress 20176	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ ∩ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
14	10, 4, 13	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
15	14	oveq1d 7463	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
16		difssd 4160	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ (Base‘𝐿))
17		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	17	subrgss 20600	. . . . . . . 8 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → ∩ 𝑆 ⊆ (Base‘𝑅))
19	5, 17	ressbas2 17296	. . . . . . . 8 ⊢ (∩ 𝑆 ⊆ (Base‘𝑅) → ∩ 𝑆 = (Base‘𝐿))
20	4, 18, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑆 = (Base‘𝐿))
21	16, 20	sseqtrrd 4050	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ ∩ 𝑆)
22		ressabs 17308	. . . . . 6 ⊢ ((∩ 𝑆 ∈ (SubRing‘𝑅) ∧ ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ ∩ 𝑆) → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
23	4, 21, 22	syl2anc 583	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
24	15, 23	eqtr3d 2782	. . . 4 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
25		intiin 5082	. . . . . . . 8 ⊢ ∩ 𝑆 = ∩ 𝑠 ∈ 𝑆 𝑠
26	20, 25	eqtr3di 2795	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐿) = ∩ 𝑠 ∈ 𝑆 𝑠)
27	26	difeq1d 4148	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
28	27	oveq2d 7464	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
29		vex 3492	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
30	29	difexi 5348	. . . . . . . . 9 ⊢ (𝑠 ∖ {(0g‘𝐿)}) ∈ V
31	30	dfiin3 5993	. . . . . . . 8 ⊢ ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
32		iindif1 5098	. . . . . . . . 9 ⊢ (𝑆 ≠ ∅ → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
33	2, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
34	31, 33	eqtr3id 2794	. . . . . . 7 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
35	34	oveq2d 7464	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
36		difss 4159	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ⊆ (Base‘𝑅)
37		eqid 2740	. . . . . . . . . . 11 ⊢ ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))
38	12, 17	mgpbas 20167	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
39	37, 38	ressbas2 17296	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ∖ {(0g‘𝑅)}) ⊆ (Base‘𝑅) → ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
40	36, 39	ax-mp 5	. . . . . . . . 9 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})))
41	40	fvexi 6934	. . . . . . . 8 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ V
42		iinssiun 5028	. . . . . . . . . . 11 ⊢ (𝑆 ≠ ∅ → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}))
43	2, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}))
44		subrgsubg 20605	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
45	44	ssriv 4012	. . . . . . . . . . . . . . . . . 18 ⊢ (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
46	1, 45	sstrdi 4021	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑅))
47		subgint 19190	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubGrp‘𝑅))
48	46, 2, 47	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubGrp‘𝑅))
49		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	5, 49	subg0 19172	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑆 ∈ (SubGrp‘𝑅) → (0g‘𝑅) = (0g‘𝐿))
51	48, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐿))
52	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0g‘𝑅) = (0g‘𝐿))
53	52	sneqd 4660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘𝐿)})
54	53	difeq2d 4149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = (𝑠 ∖ {(0g‘𝐿)}))
55	1	sselda 4008	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (SubRing‘𝑅))
56	17	subrgss 20600	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
57	55, 56	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ⊆ (Base‘𝑅))
58	57	ssdifd 4168	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
59	54, 58	eqsstrrd 4048	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	59	iunssd 5073	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
61	43, 60	sstrd 4019	. . . . . . . . 9 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
62	31, 61	eqsstrrid 4058	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
63		ressabs 17308	. . . . . . . 8 ⊢ ((((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ V ∧ ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))))
64	41, 62, 63	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))))
65	17, 49, 37	drngmgp 20767	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
66	10, 65	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
67	66	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
68	59, 40	sseqtrdi 4059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
69		ressabs 17308	. . . . . . . . . . . . . 14 ⊢ ((((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ V ∧ (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)})) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
70	41, 59, 69	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
71		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑠)
72	71, 12	mgpress 20176	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
73	10, 72	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
74	54	eqcomd 2746	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) = (𝑠 ∖ {(0g‘𝑅)}))
75	73, 74	oveq12d 7466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})))
76		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
77		difssd 4160	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ 𝑠)
78		ressabs 17308	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ 𝑆 ∧ (𝑠 ∖ {(0g‘𝐿)}) ⊆ 𝑠) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
79	76, 77, 78	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
80	75, 79	eqtr3d 2782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
81	71, 17	ressbas2 17296	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ⊆ (Base‘𝑅) → 𝑠 = (Base‘(𝑅 ↾s 𝑠)))
82	55, 56, 81	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 = (Base‘(𝑅 ↾s 𝑠)))
83	71, 49	subrg0 20607	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝑠)))
84	55, 83	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝑠)))
85	84	sneqd 4660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘(𝑅 ↾s 𝑠))})
86	82, 85	difeq12d 4150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))}))
87	86	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) = ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))})))
88		subdrgint.5	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑅 ↾s 𝑠) ∈ DivRing)
89		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑅 ↾s 𝑠)) = (Base‘(𝑅 ↾s 𝑠))
90		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑅 ↾s 𝑠)) = (0g‘(𝑅 ↾s 𝑠))
91		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))})) = ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))}))
92	89, 90, 91	drngmgp 20767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ↾s 𝑠) ∈ DivRing → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))})) ∈ Grp)
93	88, 92	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))})) ∈ Grp)
94	87, 93	eqeltrd 2844	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) ∈ Grp)
95	80, 94	eqeltrrd 2845	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
96	70, 95	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
97		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})))
98	97	issubg 19166	. . . . . . . . . . . 12 ⊢ ((𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) ↔ (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp ∧ (𝑠 ∖ {(0g‘𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) ∧ (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp))
99	67, 68, 96, 98	syl3anbrc 1343	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
100	99	ralrimiva 3152	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
101		eqid 2740	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
102	101	rnmptss 7157	. . . . . . . . . 10 ⊢ (∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) → ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
103	100, 102	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
104		dmmptg 6273	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ V → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = 𝑆)
105		difexg 5347	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝑆 → (𝑠 ∖ {(0g‘𝐿)}) ∈ V)
106	104, 105	mprg 3073	. . . . . . . . . . . 12 ⊢ dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = 𝑆
107	106	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = 𝑆)
108	107, 2	eqnetrd 3014	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
109		dm0rn0 5949	. . . . . . . . . . 11 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅)
110	109	necon3bii 2999	. . . . . . . . . 10 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
111	108, 110	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
112		subgint 19190	. . . . . . . . 9 ⊢ ((ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) ∧ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅) → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
113	103, 111, 112	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
114		eqid 2740	. . . . . . . . 9 ⊢ (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})))
115	114	subggrp 19169	. . . . . . . 8 ⊢ (∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
116	113, 115	syl 17	. . . . . . 7 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
117	64, 116	eqeltrrd 2845	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
118	35, 117	eqeltrrd 2845	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
119	28, 118	eqeltrd 2844	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
120	24, 119	eqeltrd 2844	. . 3 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
121	9, 120	eqeltrid 2848	. 2 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
122		eqid 2740	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
123		eqid 2740	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
124		eqid 2740	. . 3 ⊢ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)}))
125	122, 123, 124	isdrng2 20765	. 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp))
126	7, 121, 125	sylanbrc 582	1 ⊢ (𝜑 → 𝐿 ∈ DivRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∩ cint 4970  ∪ ciun 5015  ∩ ciin 5016   ↦ cmpt 5249  dom cdm 5700  ran crn 5701  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  0_gc0g 17499  Grpcgrp 18973  SubGrpcsubg 19160  mulGrpcmgp 20161  Ringcrg 20260  SubRingcsubrg 20595  DivRingcdr 20751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-subg 19163  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-subrng 20572  df-subrg 20597  df-drng 20753

This theorem is referenced by:  sdrgint  20827  primefld  20828  fldgensdrg  33281