Theorem subdrgint 20782

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 20574	. . . 4 ⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅))
4	1, 2, 3	syl2anc 590	. . 3 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubRing‘𝑅))
5		subdrgint.1	. . . 4 ⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)
6	5	subrgring 20553	. . 3 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → 𝐿 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝐿 ∈ Ring)
8	5	fveq2i 6837	. . . 4 ⊢ (mulGrp‘𝐿) = (mulGrp‘(𝑅 ↾s ∩ 𝑆))
9	8	oveq1i 7373	. . 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 20129	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ ∩ 𝑆 ∈ (SubRing‘𝑅)) → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
14	10, 4, 13	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ 𝑆) = (mulGrp‘(𝑅 ↾s ∩ 𝑆)))
15	14	oveq1d 7378	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
16		difssd 4074	. . . . . . 7 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ (Base‘𝐿))
17		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	17	subrgss 20551	. . . . . . . 8 ⊢ (∩ 𝑆 ∈ (SubRing‘𝑅) → ∩ 𝑆 ⊆ (Base‘𝑅))
19	5, 17	ressbas2 17206	. . . . . . . 8 ⊢ (∩ 𝑆 ⊆ (Base‘𝑅) → ∩ 𝑆 = (Base‘𝐿))
20	4, 18, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑆 = (Base‘𝐿))
21	16, 20	sseqtrrd 3959	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) ⊆ ∩ 𝑆)
22		ressabs 17216	. . . . . 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 590	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝑅) ↾s ∩ 𝑆) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
24	15, 23	eqtr3d 2777	. . . 4 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})))
25		intiin 4996	. . . . . . . 8 ⊢ ∩ 𝑆 = ∩ 𝑠 ∈ 𝑆 𝑠
26	20, 25	eqtr3di 2790	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐿) = ∩ 𝑠 ∈ 𝑆 𝑠)
27	26	difeq1d 4063	. . . . . 6 ⊢ (𝜑 → ((Base‘𝐿) ∖ {(0g‘𝐿)}) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
28	27	oveq2d 7379	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
29		vex 3436	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
30	29	difexi 5265	. . . . . . . . 9 ⊢ (𝑠 ∖ {(0g‘𝐿)}) ∈ V
31	30	dfiin3 5920	. . . . . . . 8 ⊢ ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) = ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
32		iindif1 5011	. . . . . . . . 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 2789	. . . . . . 7 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)}))
35	34	oveq2d 7379	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) = ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})))
36		difss 4073	. . . . . . . . . 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 20124	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
39	37, 38	ressbas2 17206	. . . . . . . . . 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 6848	. . . . . . . 8 ⊢ ((Base‘𝑅) ∖ {(0g‘𝑅)}) ∈ V
42		iinssiun 4942	. . . . . . . . . . 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 20556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ∈ (SubGrp‘𝑅))
45	44	ssriv 3926	. . . . . . . . . . . . . . . . . 18 ⊢ (SubRing‘𝑅) ⊆ (SubGrp‘𝑅)
46	1, 45	sstrdi 3934	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑅))
47		subgint 19124	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ (SubGrp‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubGrp‘𝑅))
48	46, 2, 47	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∩ 𝑆 ∈ (SubGrp‘𝑅))
49		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	5, 49	subg0 19106	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑆 ∈ (SubGrp‘𝑅) → (0g‘𝑅) = (0g‘𝐿))
51	48, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐿))
52	51	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0g‘𝑅) = (0g‘𝐿))
53	52	sneqd 4574	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘𝐿)})
54	53	difeq2d 4064	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = (𝑠 ∖ {(0g‘𝐿)}))
55	1	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (SubRing‘𝑅))
56	17	subrgss 20551	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (SubRing‘𝑅) → 𝑠 ⊆ (Base‘𝑅))
57	55, 56	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ⊆ (Base‘𝑅))
58	57	ssdifd 4082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
59	54, 58	eqsstrrd 3957	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
60	59	iunssd 4987	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
61	43, 60	sstrd 3932	. . . . . . . . 9 ⊢ (𝜑 → ∩ 𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
62	31, 61	eqsstrrid 3961	. . . . . . . 8 ⊢ (𝜑 → ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ⊆ ((Base‘𝑅) ∖ {(0g‘𝑅)}))
63		ressabs 17216	. . . . . . . 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 593	. . . . . . 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 20724	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)})) ∈ Grp)
68	59, 40	sseqtrdi 3962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ (Base‘((mulGrp‘𝑅) ↾s ((Base‘𝑅) ∖ {(0g‘𝑅)}))))
69		ressabs 17216	. . . . . . . . . . . . . 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 593	. . . . . . . . . . . . 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 20129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s 𝑠) = (mulGrp‘(𝑅 ↾s 𝑠)))
73	10, 72	sylan 586	. . . . . . . . . . . . . . . 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 7381	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})))
76		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
77		difssd 4074	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝐿)}) ⊆ 𝑠)
78		ressabs 17216	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((mulGrp‘𝑅) ↾s 𝑠) ↾s (𝑠 ∖ {(0g‘𝐿)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
80	75, 79	eqtr3d 2777	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) = ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})))
81	71, 17	ressbas2 17206	. . . . . . . . . . . . . . . . . 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 20558	. . . . . . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(0g‘𝑅)} = {(0g‘(𝑅 ↾s 𝑠))})
86	82, 85	difeq12d 4065	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∖ {(0g‘𝑅)}) = ((Base‘(𝑅 ↾s 𝑠)) ∖ {(0g‘(𝑅 ↾s 𝑠))}))
87	86	oveq2d 7379	. . . . . . . . . . . . . . 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 20724	. . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘(𝑅 ↾s 𝑠)) ↾s (𝑠 ∖ {(0g‘𝑅)})) ∈ Grp)
95	80, 94	eqeltrrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((mulGrp‘𝑅) ↾s (𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
96	70, 95	eqeltrd 2840	. . . . . . . . . . . 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 19100	. . . . . . . . . . . 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 1350	. . . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))
102	101	rnmptss 7071	. . . . . . . . . 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 6200	. . . . . . . . . . . . 13 ⊢ (∀𝑠 ∈ 𝑆 (𝑠 ∖ {(0g‘𝐿)}) ∈ V → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = 𝑆)
105		difexg 5264	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝑆 → (𝑠 ∖ {(0g‘𝐿)}) ∈ V)
106	104, 105	mprg 3060	. . . . . . . . . . . 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 3002	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
109		dm0rn0 5873	. . . . . . . . . . 11 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) = ∅)
110	109	necon3bii 2987	. . . . . . . . . 10 ⊢ (dom (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅ ↔ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
111	108, 110	sylib 219	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)})) ≠ ∅)
112		subgint 19124	. . . . . . . . 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 590	. . . . . . . 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 19103	. . . . . . . 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 2841	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ∩ ran (𝑠 ∈ 𝑆 ↦ (𝑠 ∖ {(0g‘𝐿)}))) ∈ Grp)
118	35, 117	eqeltrrd 2841	. . . . 5 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s (∩ 𝑠 ∈ 𝑆 𝑠 ∖ {(0g‘𝐿)})) ∈ Grp)
119	28, 118	eqeltrd 2840	. . . 4 ⊢ (𝜑 → ((mulGrp‘𝑅) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
120	24, 119	eqeltrd 2840	. . 3 ⊢ (𝜑 → ((mulGrp‘(𝑅 ↾s ∩ 𝑆)) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp)
121	9, 120	eqeltrid 2844	. 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 20722	. 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ ((mulGrp‘𝐿) ↾s ((Base‘𝐿) ∖ {(0g‘𝐿)})) ∈ Grp))
126	7, 121, 125	sylanbrc 589	1 ⊢ (𝜑 → 𝐿 ∈ DivRing)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4268  {csn 4562  ∩ cint 4884  ∪ ciun 4928  ∩ ciin 4929   ↦ cmpt 5160  dom cdm 5625  ran crn 5626  ‘cfv 6492  (class class class)co 7363  Basecbs 17177   ↾_s cress 17198  0_gc0g 17400  Grpcgrp 18907  SubGrpcsubg 19094  mulGrpcmgp 20119  Ringcrg 20212  SubRingcsubrg 20548  DivRingcdr 20708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-tpos 8173  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-subg 19097  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20315  df-dvdsr 20335  df-unit 20336  df-invr 20366  df-dvr 20379  df-subrng 20525  df-subrg 20549  df-drng 20710

This theorem is referenced by:  sdrgint  20783  primefld  20784  fldgensdrg  33405