Theorem subgintm 13609

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 3913	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝑆 → ∩ 𝑆 ⊆ ∪ 𝑆)
2	1	adantl 277	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ ∪ 𝑆)
3		ssel2 3192	. . . . . . 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 2206	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	subgss 13585	. . . . . 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 2580	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 𝑔 ⊆ (Base‘𝐺))
9		unissb 3886	. . . 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 3207	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ (Base‘𝐺))
12		eqid 2206	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	12	subg0cl 13593	. . . . . 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 2580	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 (0g‘𝐺) ∈ 𝑔)
16		ssel 3191	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubGrp‘𝐺) → (𝑤 ∈ 𝑆 → 𝑤 ∈ (SubGrp‘𝐺)))
17	16	eximdv 1904	. . . . . . 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 13590	. . . . . . 7 ⊢ (𝑤 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
20	19	exlimiv 1622	. . . . . 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 13436	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
23		elintg 3899	. . . . 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 2790	. . 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 3900	. . . . . . . . . . 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 3900	. . . . . . . . . . 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 2206	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
38	37	subgcl 13595	. . . . . . . . 9 ⊢ ((𝑔 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑔 ∧ 𝑦 ∈ 𝑔) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
39	28, 32, 36, 38	syl3anc 1250	. . . . . . . 8 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
40	39	ralrimiva 2580	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
41		vex 2776	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑥 ∈ V)
43		plusgslid 13019	. . . . . . . . . . . 12 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
44	43	slotex 12934	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) ∈ V)
45	18, 20, 44	3syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (+g‘𝐺) ∈ V)
46		vex 2776	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	46	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑦 ∈ V)
48		ovexg 5991	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑦 ∈ V) → (𝑥(+g‘𝐺)𝑦) ∈ V)
49	42, 45, 47, 48	syl3anc 1250	. . . . . . . . 9 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ V)
50		elintg 3899	. . . . . . . . 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 2580	. . . 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 2206	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
59	58	subginvcl 13594	. . . . . . 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 2580	. . . . 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 3197	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → 𝑥 ∈ (Base‘𝐺))
64	5, 58	grpinvcl 13455	. . . . . . 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 3899	. . . . . 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 2580	. 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 13600	. . 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 1183	1 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  Vcvv 2773   ⊆ wss 3170  ∪ cuni 3856  ∩ cint 3891  ‘cfv 5280  (class class class)co 5957  Basecbs 12907  +_gcplusg 12984  0_gc0g 13163  Grpcgrp 13407  inv_gcminusg 13408  SubGrpcsubg 13578

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-subg 13581

This theorem is referenced by:  subrngintm  14049  subrgintm  14080