Theorem subdrgint 20763

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 20555	. . . 4 ⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅))
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubRing‘𝑅))
5		subdrgint.1	. . . 4 ⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)
6	5	subrgring 20534	. . 3 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐿 ∈ Ring)
8	5	fveq2i 6879	. . . 4 ⊢ (mulGrp‘𝐿) = (mulGrp‘(𝑅 ↾s ∩ 𝑆))
9	8	oveq1i 7415	. . 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 2735	. . . . . . . 8 ⊢ (𝑅 ↾s ∩ 𝑆) = (𝑅 ↾s ∩ 𝑆)
12		eqid 2735	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
13	11, 12	mgpress 20110	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ ∩ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
14	10, 4, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
15	14	oveq1d 7420	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
16		difssd 4112	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ (Base‘𝐿))
17		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	17	subrgss 20532	. . . . . . . 8 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → ∩ 𝑆 ⊆ (Base‘𝑅))
19	5, 17	ressbas2 17259	. . . . . . . 8 ⊢ (∩ 𝑆 ⊆ (Base‘𝑅) → ∩ 𝑆 = (Base‘𝐿))
20	4, 18, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑆 = (Base‘𝐿))
21	16, 20	sseqtrrd 3996	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ ∩ 𝑆)
22		ressabs 17269	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
24	15, 23	eqtr3d 2772	. . . 4 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
25		intiin 5035	. . . . . . . 8 ⊢ ∩ 𝑆 = ∩ 𝑠 ∈ 𝑆 𝑠
26	20, 25	eqtr3di 2785	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐿) = ∩ 𝑠 ∈ 𝑆 𝑠)
27	26	difeq1d 4100	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
28	27	oveq2d 7421	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
29		vex 3463	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
30	29	difexi 5300	. . . . . . . . 9 ⊢ (𝑠 ∖ {(0g‘𝐿)}) ∈ V
31	30	dfiin3 5950	. . . . . . . 8 ⊢ ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
32		iindif1 5051	. . . . . . . . 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 2784	. . . . . . 7 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
35	34	oveq2d 7421	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
36		difss 4111	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ⊆ (Base‘𝑅)
37		eqid 2735	. . . . . . . . . . 11 ⊢ ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))
38	12, 17	mgpbas 20105	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
39	37, 38	ressbas2 17259	. . . . . . . . . 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 6890	. . . . . . . 8 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ V
42		iinssiun 4981	. . . . . . . . . . 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 20537	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
45	44	ssriv 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
46	1, 45	sstrdi 3971	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑅))
47		subgint 19133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubGrp‘𝑅))
48	46, 2, 47	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubGrp‘𝑅))
49		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	5, 49	subg0 19115	. . . . . . . . . . . . . . . 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 4613	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘𝐿)})
54	53	difeq2d 4101	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = (𝑠 ∖ {(0g‘𝐿)}))
55	1	sselda 3958	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (SubRing‘𝑅))
56	17	subrgss 20532	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
57	55, 56	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ⊆ (Base‘𝑅))
58	57	ssdifd 4120	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
59	54, 58	eqsstrrd 3994	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	59	iunssd 5026	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
61	43, 60	sstrd 3969	. . . . . . . . 9 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
62	31, 61	eqsstrrid 3998	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
63		ressabs 17269	. . . . . . . 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 587	. . . . . . 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 20705	. . . . . . . . . . . . . 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 3999	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
69		ressabs 17269	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
71		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑠)
72	71, 12	mgpress 20110	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
73	10, 72	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
74	54	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) = (𝑠 ∖ {(0g‘𝑅)}))
75	73, 74	oveq12d 7423	. . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ 𝑠)
78		ressabs 17269	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
80	75, 79	eqtr3d 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
81	71, 17	ressbas2 17259	. . . . . . . . . . . . . . . . . 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 20539	. . . . . . . . . . . . . . . . . . 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 4613	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘(𝑅 ↾s 𝑠))})
86	82, 85	difeq12d 4102	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))}))
87	86	oveq2d 7421	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑅 ↾s 𝑠)) = (Base‘(𝑅 ↾s 𝑠))
90		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘(𝑅 ↾s 𝑠)) = (0g‘(𝑅 ↾s 𝑠))
91		eqid 2735	. . . . . . . . . . . . . . . . 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 20705	. . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) ∈ Grp)
95	80, 94	eqeltrrd 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
96	70, 95	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
97		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))) = (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})))
98	97	issubg 19109	. . . . . . . . . . . 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 1344	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
100	99	ralrimiva 3132	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
101		eqid 2735	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
102	101	rnmptss 7113	. . . . . . . . . 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 6231	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ V → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = 𝑆)
105		difexg 5299	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝑆 → (𝑠 ∖ {(0g‘𝐿)}) ∈ V)
106	104, 105	mprg 3057	. . . . . . . . . . . 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 2999	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
109		dm0rn0 5904	. . . . . . . . . . 11 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅)
110	109	necon3bii 2984	. . . . . . . . . 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 19133	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ∈ (SubGrp‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
114		eqid 2735	. . . . . . . . 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 19112	. . . . . . . 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 2835	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
118	35, 117	eqeltrrd 2835	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
119	28, 118	eqeltrd 2834	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
120	24, 119	eqeltrd 2834	. . 3 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
121	9, 120	eqeltrid 2838	. 2 ⊢ (𝜑 → ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
122		eqid 2735	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
123		eqid 2735	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
124		eqid 2735	. . 3 ⊢ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)}))
125	122, 123, 124	isdrng2 20703	. 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp))
126	7, 121, 125	sylanbrc 583	1 ⊢ (𝜑 → 𝐿 ∈ DivRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  {csn 4601  ∩ cint 4922  ∪ ciun 4967  ∩ ciin 4968   ↦ cmpt 5201  dom cdm 5654  ran crn 5655  ‘cfv 6531  (class class class)co 7405  Basecbs 17228   ↾_s cress 17251  0_gc0g 17453  Grpcgrp 18916  SubGrpcsubg 19103  mulGrpcmgp 20100  Ringcrg 20193  SubRingcsubrg 20529  DivRingcdr 20689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-subg 19106  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-dvr 20361  df-subrng 20506  df-subrg 20530  df-drng 20691

This theorem is referenced by:  sdrgint  20764  primefld  20765  fldgensdrg  33308