Theorem subgintm 13404

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 3897	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝑆 → ∩ 𝑆 ⊆ ∪ 𝑆)
2	1	adantl 277	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ ∪ 𝑆)
3		ssel2 3179	. . . . . . 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 2196	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	subgss 13380	. . . . . 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 2570	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 𝑔 ⊆ (Base‘𝐺))
9		unissb 3870	. . . 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 3194	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ (Base‘𝐺))
12		eqid 2196	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	12	subg0cl 13388	. . . . . 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 2570	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 (0g‘𝐺) ∈ 𝑔)
16		ssel 3178	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubGrp‘𝐺) → (𝑤 ∈ 𝑆 → 𝑤 ∈ (SubGrp‘𝐺)))
17	16	eximdv 1894	. . . . . . 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 13385	. . . . . . 7 ⊢ (𝑤 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
20	19	exlimiv 1612	. . . . . 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 13231	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
23		elintg 3883	. . . . 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 2779	. . 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 529	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑥 ∈ ∩ 𝑆)
30		elinti 3884	. . . . . . . . . . 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 531	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → 𝑦 ∈ ∩ 𝑆)
34		elinti 3884	. . . . . . . . . . 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 2196	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
38	37	subgcl 13390	. . . . . . . . 9 ⊢ ((𝑔 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑔 ∧ 𝑦 ∈ 𝑔) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
39	28, 32, 36, 38	syl3anc 1249	. . . . . . . 8 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
40	39	ralrimiva 2570	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
41		vex 2766	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑥 ∈ V)
43		plusgslid 12815	. . . . . . . . . . . 12 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
44	43	slotex 12730	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) ∈ V)
45	18, 20, 44	3syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (+g‘𝐺) ∈ V)
46		vex 2766	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	46	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑦 ∈ V)
48		ovexg 5959	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑦 ∈ V) → (𝑥(+g‘𝐺)𝑦) ∈ V)
49	42, 45, 47, 48	syl3anc 1249	. . . . . . . . 9 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ V)
50		elintg 3883	. . . . . . . . 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 2570	. . . 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 2196	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
59	58	subginvcl 13389	. . . . . . 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 2570	. . . . 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 3184	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → 𝑥 ∈ (Base‘𝐺))
64	5, 58	grpinvcl 13250	. . . . . . 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 3883	. . . . . 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 2570	. 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 13395	. . 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 1182	1 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157  ∪ cuni 3840  ∩ cint 3875  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  Grpcgrp 13202  inv_gcminusg 13203  SubGrpcsubg 13373

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-subg 13376

This theorem is referenced by:  subrngintm  13844  subrgintm  13875