Theorem subgintm 13730

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 3944	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝑆 → ∩ 𝑆 ⊆ ∪ 𝑆)
2	1	adantl 277	. . 3 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ ∪ 𝑆)
3		ssel2 3219	. . . . . . 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 2229	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5	subgss 13706	. . . . . 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 2603	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 𝑔 ⊆ (Base‘𝐺))
9		unissb 3917	. . . 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 3234	. 2 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ⊆ (Base‘𝐺))
12		eqid 2229	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	12	subg0cl 13714	. . . . . 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 2603	. . . 4 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∀𝑔 ∈ 𝑆 (0g‘𝐺) ∈ 𝑔)
16		ssel 3218	. . . . . . . 8 ⊢ (𝑆 ⊆ (SubGrp‘𝐺) → (𝑤 ∈ 𝑆 → 𝑤 ∈ (SubGrp‘𝐺)))
17	16	eximdv 1926	. . . . . . 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 13711	. . . . . . 7 ⊢ (𝑤 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
20	19	exlimiv 1644	. . . . . 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 13557	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
23		elintg 3930	. . . . 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 2816	. . 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 3931	. . . . . . . . . . 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 3931	. . . . . . . . . . 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 2229	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
38	37	subgcl 13716	. . . . . . . . 9 ⊢ ((𝑔 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑔 ∧ 𝑦 ∈ 𝑔) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
39	28, 32, 36, 38	syl3anc 1271	. . . . . . . 8 ⊢ ((((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
40	39	ralrimiva 2603	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) → ∀𝑔 ∈ 𝑆 (𝑥(+g‘𝐺)𝑦) ∈ 𝑔)
41		vex 2802	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑥 ∈ V)
43		plusgslid 13140	. . . . . . . . . . . 12 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
44	43	slotex 13054	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (+g‘𝐺) ∈ V)
45	18, 20, 44	3syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (+g‘𝐺) ∈ V)
46		vex 2802	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
47	46	a1i 9	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → 𝑦 ∈ V)
48		ovexg 6034	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑦 ∈ V) → (𝑥(+g‘𝐺)𝑦) ∈ V)
49	42, 45, 47, 48	syl3anc 1271	. . . . . . . . 9 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ V)
50		elintg 3930	. . . . . . . . 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 2603	. . . 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 2229	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
59	58	subginvcl 13715	. . . . . . 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 2603	. . . . 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 3224	. . . . . . 7 ⊢ (((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) ∧ 𝑥 ∈ ∩ 𝑆) → 𝑥 ∈ (Base‘𝐺))
64	5, 58	grpinvcl 13576	. . . . . . 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 3930	. . . . . 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 2603	. 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 13721	. . 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 1204	1 ⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197  ∪ cuni 3887  ∩ cint 3922  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284  Grpcgrp 13528  inv_gcminusg 13529  SubGrpcsubg 13699

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-subg 13702

This theorem is referenced by:  subrngintm  14170  subrgintm  14201