Theorem subdrgint 20887

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 20681	. . . 4 ⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅))
4	1, 2, 3	syl2anc 595	. . 3 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubRing‘𝑅))
5		subdrgint.1	. . . 4 ⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)
6	5	subrgring 20660	. . 3 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
7	4, 6	syl 18	. 2 ⊢ (𝜑 → 𝐿 ∈ Ring)
8	5	fveq2i 6886	. . . 4 ⊢ (mulGrp‘𝐿) = (mulGrp‘(𝑅 ↾s ∩ 𝑆))
9	8	oveq1i 7422	. . 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 2763	. . . . . . . 8 ⊢ (𝑅 ↾s ∩ 𝑆) = (𝑅 ↾s ∩ 𝑆)
12		eqid 2763	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
13	11, 12	mgpress 20227	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ ∩ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
14	10, 4, 13	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
15	14	oveq1d 7427	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
16		difssd 4092	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ (Base‘𝐿))
17		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	17	subrgss 20658	. . . . . . . 8 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → ∩ 𝑆 ⊆ (Base‘𝑅))
19	5, 17	ressbas2 17299	. . . . . . . 8 ⊢ (∩ 𝑆 ⊆ (Base‘𝑅) → ∩ 𝑆 = (Base‘𝐿))
20	4, 18, 19	3syl 19	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑆 = (Base‘𝐿))
21	16, 20	sseqtrrd 3975	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ ∩ 𝑆)
22		ressabs 17309	. . . . . 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 595	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
24	15, 23	eqtr3d 2800	. . . 4 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
25		intiin 5025	. . . . . . . 8 ⊢ ∩ 𝑆 = ∩ 𝑠 ∈ 𝑆 𝑠
26	20, 25	eqtr3di 2813	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐿) = ∩ 𝑠 ∈ 𝑆 𝑠)
27	26	difeq1d 4081	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
28	27	oveq2d 7428	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
29		vex 3459	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
30	29	difexi 5302	. . . . . . . . 9 ⊢ (𝑠 ∖ {(0g‘𝐿)}) ∈ V
31	30	dfiin3 5963	. . . . . . . 8 ⊢ ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
32		iindif1 5042	. . . . . . . . 9 ⊢ (𝑆 ≠ ∅ → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
33	2, 32	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
34	31, 33	eqtr3id 2812	. . . . . . 7 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
35	34	oveq2d 7428	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
36		difss 4091	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ⊆ (Base‘𝑅)
37		eqid 2763	. . . . . . . . . . 11 ⊢ ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))
38	12, 17	mgpbas 20222	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
39	37, 38	ressbas2 17299	. . . . . . . . . 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 6897	. . . . . . . 8 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ V
42		iinssiun 4971	. . . . . . . . . . 11 ⊢ (𝑆 ≠ ∅ → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}))
43	2, 42	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}))
44		subrgsubg 20663	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
45	44	ssriv 3942	. . . . . . . . . . . . . . . . . 18 ⊢ (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
46	1, 45	sstrdi 3950	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑅))
47		subgint 19218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubGrp‘𝑅))
48	46, 2, 47	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubGrp‘𝑅))
49		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	5, 49	subg0 19199	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑆 ∈ (SubGrp‘𝑅) → (0g‘𝑅) = (0g‘𝐿))
51	48, 50	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐿))
52	51	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0g‘𝑅) = (0g‘𝐿))
53	52	sneqd 4602	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘𝐿)})
54	53	difeq2d 4082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = (𝑠 ∖ {(0g‘𝐿)}))
55	1	sselda 3938	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (SubRing‘𝑅))
56	17	subrgss 20658	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
57	55, 56	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ⊆ (Base‘𝑅))
58	57	ssdifd 4100	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
59	54, 58	eqsstrrd 3973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	59	iunssd 5016	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
61	43, 60	sstrd 3948	. . . . . . . . 9 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
62	31, 61	eqsstrrid 3977	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
63		ressabs 17309	. . . . . . . 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 598	. . . . . . 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 20832	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
66	10, 65	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
67	66	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
68	59, 40	sseqtrdi 3978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
69		ressabs 17309	. . . . . . . . . . . . . 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 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
71		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑠)
72	71, 12	mgpress 20227	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
73	10, 72	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
74	54	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) = (𝑠 ∖ {(0g‘𝑅)}))
75	73, 74	oveq12d 7430	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})))
76		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
77		difssd 4092	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ 𝑠)
78		ressabs 17309	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
80	75, 79	eqtr3d 2800	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
81	71, 17	ressbas2 17299	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ⊆ (Base‘𝑅) → 𝑠 = (Base‘(𝑅 ↾s 𝑠)))
82	55, 56, 81	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 = (Base‘(𝑅 ↾s 𝑠)))
83	71, 49	subrg0 20665	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝑠)))
84	55, 83	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0g‘𝑅) = (0g‘(𝑅 ↾s 𝑠)))
85	84	sneqd 4602	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘(𝑅 ↾s 𝑠))})
86	82, 85	difeq12d 4083	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))}))
87	86	oveq2d 7428	. . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑅 ↾s 𝑠)) = (Base‘(𝑅 ↾s 𝑠))
90		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑅 ↾s 𝑠)) = (0g‘(𝑅 ↾s 𝑠))
91		eqid 2763	. . . . . . . . . . . . . . . . 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 20832	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ↾s 𝑠) ∈ DivRing → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))})) ∈ Grp)
93	88, 92	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))})) ∈ Grp)
94	87, 93	eqeltrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) ∈ Grp)
95	80, 94	eqeltrrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
96	70, 95	eqeltrd 2863	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
97		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})))
98	97	issubg 19193	. . . . . . . . . . . 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 1362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
100	99	ralrimiva 3157	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
101		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
102	101	rnmptss 7120	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
104		dmmptg 6245	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ V → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = 𝑆)
105		difexg 5301	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝑆 → (𝑠 ∖ {(0g‘𝐿)}) ∈ V)
106	104, 105	mprg 3085	. . . . . . . . . . . 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 3025	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
109		dm0rn0 5916	. . . . . . . . . . 11 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅)
110	109	necon3bii 3010	. . . . . . . . . 10 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
111	108, 110	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
112		subgint 19218	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
114		eqid 2763	. . . . . . . . 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 19196	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
117	64, 116	eqeltrrd 2864	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
118	35, 117	eqeltrrd 2864	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
119	28, 118	eqeltrd 2863	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
120	24, 119	eqeltrd 2863	. . 3 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
121	9, 120	eqeltrid 2867	. 2 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
122		eqid 2763	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
123		eqid 2763	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
124		eqid 2763	. . 3 ⊢ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)}))
125	122, 123, 124	isdrng2 20830	. 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp))
126	7, 121, 125	sylanbrc 594	1 ⊢ (𝜑 → 𝐿 ∈ DivRing)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  {csn 4590  ∩ cint 4913  ∪ ciun 4957  ∩ ciin 4958   ↦ cmpt 5193  dom cdm 5663  ran crn 5664  ‘cfv 6538  (class class class)co 7412  Basecbs 17270   ↾_s cress 17291  0_gc0g 17493  Grpcgrp 19001  SubGrpcsubg 19187  mulGrpcmgp 20217  Ringcrg 20316  SubRingcsubrg 20655  DivRingcdr 20814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-tpos 8223  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-mulr 17325  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-minusg 19005  df-subg 19190  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-oppr 20420  df-dvdsr 20440  df-unit 20441  df-invr 20471  df-dvr 20484  df-subrng 20632  df-subrg 20656  df-drng 20816

This theorem is referenced by:  sdrgint  20888  primefld  20889  fldgensdrg  33613