Theorem subgintm 13848

Description: The intersection of an inhabited collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)

Assertion

Ref	Expression
subgintm	⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝐺))

Distinct variable groups:   𝑤,𝐺   𝑤,𝑆

Proof of Theorem subgintm

Dummy variables 𝑥 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		intssunim 3955	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝑆 → ∩ 𝑆 ⊆ ∪ 𝑆)
2	1	adantl 277	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ ∪ 𝑆)
3		ssel2 3223	. . . . . . 7 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ 𝑔 ∈ 𝑆) → 𝑔 ∈ (SubGrp‘𝐺))
4	3	adantlr 477	. . . . . 6 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑔 ∈ 𝑆) → 𝑔 ∈ (SubGrp‘𝐺))
5		eqid 2231	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	subgss 13824	. . . . . 6 ⊢ (𝑔 ∈ (SubGrp‘𝐺) → 𝑔 ⊆ (Base‘𝐺))
7	4, 6	syl 14	. . . . 5 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑔 ∈ 𝑆) → 𝑔 ⊆ (Base‘𝐺))
8	7	ralrimiva 2606	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 𝑔 ⊆ (Base‘𝐺))
9		unissb 3928	. . . 4 ⊢ (∪ 𝑆 ⊆ (Base‘𝐺) ↔ ∀𝑔 ∈ 𝑆 𝑔 ⊆ (Base‘𝐺))
10	8, 9	sylibr 134	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∪ 𝑆 ⊆ (Base‘𝐺))
11	2, 10	sstrd 3238	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ (Base‘𝐺))
12		eqid 2231	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	12	subg0cl 13832	. . . . . 6 ⊢ (𝑔 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑔)
14	4, 13	syl 14	. . . . 5 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑔 ∈ 𝑆) → (0g‘𝐺) ∈ 𝑔)
15	14	ralrimiva 2606	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 (0g‘𝐺) ∈ 𝑔)
16		ssel 3222	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubGrp‘𝐺) → (𝑤 ∈ 𝑆 → 𝑤 ∈ (SubGrp‘𝐺)))
17	16	eximdv 1928	. . . . . . 7 ⊢ (𝑆 ⊆ (SubGrp‘𝐺) → (∃𝑤 𝑤 ∈ 𝑆 → ∃𝑤 𝑤 ∈ (SubGrp‘𝐺)))
18	17	imp 124	. . . . . 6 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∃𝑤 𝑤 ∈ (SubGrp‘𝐺))
19		subgrcl 13829	. . . . . . 7 ⊢ (𝑤 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
20	19	exlimiv 1647	. . . . . 6 ⊢ (∃𝑤 𝑤 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21	18, 20	syl 14	. . . . 5 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝐺 ∈ Grp)
22	5, 12	grpidcl 13675	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
23		elintg 3941	. . . . 5 ⊢ ((0g‘𝐺) ∈ (Base‘𝐺) → ((0g‘𝐺) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 (0g‘𝐺) ∈ 𝑔))
24	21, 22, 23	3syl 17	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ((0g‘𝐺) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 (0g‘𝐺) ∈ 𝑔))
25	15, 24	mpbird 167	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (0g‘𝐺) ∈ ∩ 𝑆)
26		elex2 2820	. . 3 ⊢ ((0g‘𝐺) ∈ ∩ 𝑆 → ∃𝑤 𝑤 ∈ ∩ 𝑆)
27	25, 26	syl 14	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∃𝑤 𝑤 ∈ ∩ 𝑆)
28	4	adantlr 477	. . . . . . . . 9 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → 𝑔 ∈ (SubGrp‘𝐺))
29		simprl 531	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑥 ∈ ∩ 𝑆)
30		elinti 3942	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑆 → (𝑔 ∈ 𝑆 → 𝑥 ∈ 𝑔))
31	30	imp 124	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑆 ∧ 𝑔 ∈ 𝑆) → 𝑥 ∈ 𝑔)
32	29, 31	sylan 283	. . . . . . . . 9 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → 𝑥 ∈ 𝑔)
33		simprr 533	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑦 ∈ ∩ 𝑆)
34		elinti 3942	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∩ 𝑆 → (𝑔 ∈ 𝑆 → 𝑦 ∈ 𝑔))
35	34	imp 124	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ∩ 𝑆 ∧ 𝑔 ∈ 𝑆) → 𝑦 ∈ 𝑔)
36	33, 35	sylan 283	. . . . . . . . 9 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → 𝑦 ∈ 𝑔)
37		eqid 2231	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
38	37	subgcl 13834	. . . . . . . . 9 ⊢ ((𝑔 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑔 ∧ 𝑦 ∈ 𝑔) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
39	28, 32, 36, 38	syl3anc 1274	. . . . . . . 8 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
40	39	ralrimiva 2606	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
41		vex 2806	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑥 ∈ V)
43		plusgslid 13258	. . . . . . . . . . . 12 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
44	43	slotex 13172	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) ∈ V)
45	18, 20, 44	3syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (+g‘𝐺) ∈ V)
46		vex 2806	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	46	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑦 ∈ V)
48		ovexg 6062	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑦 ∈ V) → (𝑥(+g‘𝐺)𝑦) ∈ V)
49	42, 45, 47, 48	syl3anc 1274	. . . . . . . . 9 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ V)
50		elintg 3941	. . . . . . . . 9 ⊢ ((𝑥(+g‘𝐺)𝑦) ∈ V → ((𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔))
51	49, 50	syl 14	. . . . . . . 8 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔))
52	51	adantr 276	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ((𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔))
53	40, 52	mpbird 167	. . . . . 6 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → (𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆)
54	53	anassrs 400	. . . . 5 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) ∧ 𝑦 ∈ ∩ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆)
55	54	ralrimiva 2606	. . . 4 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → ∀𝑦 ∈ ∩ 𝑆(𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆)
56	4	adantlr 477	. . . . . . 7 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) ∧ 𝑔 ∈ 𝑆) → 𝑔 ∈ (SubGrp‘𝐺))
57	31	adantll 476	. . . . . . 7 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) ∧ 𝑔 ∈ 𝑆) → 𝑥 ∈ 𝑔)
58		eqid 2231	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
59	58	subginvcl 13833	. . . . . . 7 ⊢ ((𝑔 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑔) → ((invg‘𝐺)‘𝑥) ∈ 𝑔)
60	56, 57, 59	syl2anc 411	. . . . . 6 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) ∧ 𝑔 ∈ 𝑆) → ((invg‘𝐺)‘𝑥) ∈ 𝑔)
61	60	ralrimiva 2606	. . . . 5 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → ∀𝑔 ∈ 𝑆 ((invg‘𝐺)‘𝑥) ∈ 𝑔)
62	21	adantr 276	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → 𝐺 ∈ Grp)
63	11	sselda 3228	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → 𝑥 ∈ (Base‘𝐺))
64	5, 58	grpinvcl 13694	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
65	62, 63, 64	syl2anc 411	. . . . . 6 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
66		elintg 3941	. . . . . 6 ⊢ (((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺) → (((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 ((invg‘𝐺)‘𝑥) ∈ 𝑔))
67	65, 66	syl 14	. . . . 5 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → (((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆 ↔ ∀𝑔 ∈ 𝑆 ((invg‘𝐺)‘𝑥) ∈ 𝑔))
68	61, 67	mpbird 167	. . . 4 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → ((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆)
69	55, 68	jca 306	. . 3 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → (∀𝑦 ∈ ∩ 𝑆(𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ∧ ((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆))
70	69	ralrimiva 2606	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑥 ∈ ∩ 𝑆(∀𝑦 ∈ ∩ 𝑆(𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ∧ ((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆))
71	5, 37, 58	issubg2m 13839	. . 3 ⊢ (𝐺 ∈ Grp → (∩ 𝑆 ∈ (SubGrp‘𝐺) ↔ (∩ 𝑆 ⊆ (Base‘𝐺) ∧ ∃𝑤 𝑤 ∈ ∩ 𝑆 ∧ ∀𝑥 ∈ ∩ 𝑆(∀𝑦 ∈ ∩ 𝑆(𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ∧ ((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆))))
72	18, 20, 71	3syl 17	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (∩ 𝑆 ∈ (SubGrp‘𝐺) ↔ (∩ 𝑆 ⊆ (Base‘𝐺) ∧ ∃𝑤 𝑤 ∈ ∩ 𝑆 ∧ ∀𝑥 ∈ ∩ 𝑆(∀𝑦 ∈ ∩ 𝑆(𝑥(+g‘𝐺)𝑦) ∈ ∩ 𝑆 ∧ ((invg‘𝐺)‘𝑥) ∈ ∩ 𝑆))))
73	11, 27, 70, 72	mpbir3and 1207	1 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  ∪ cuni 3898  ∩ cint 3933  ‘cfv 5333  (class class class)co 6028  Basecbs 13145  +_gcplusg 13223  0_gc0g 13402  Grpcgrp 13646  inv_gcminusg 13647  SubGrpcsubg 13817

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-subg 13820

This theorem is referenced by:  subrngintm  14290  subrgintm  14321