Theorem issubg2m 13839

Description: Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
issubg2.b	⊢ 𝐵 = (Base‘𝐺)
issubg2.p	⊢ + = (+g‘𝐺)
issubg2.i	⊢ 𝐼 = (invg‘𝐺)

Assertion

Ref	Expression
issubg2m	⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑢 𝑢 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))))

Distinct variable groups:   𝑢, + ,𝑥,𝑦   𝑢,𝐵   𝑢,𝐺,𝑥,𝑦   𝑢,𝐼,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem issubg2m

Dummy variables 𝑣 𝑤 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issubg2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2	1	subgss 13824	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
3		eqid 2231	. . . . . . 7 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆)
4	3	subggrp 13827	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp)
5		eqid 2231	. . . . . . 7 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆))
6		eqid 2231	. . . . . . 7 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆))
7	5, 6	grpidcl 13675	. . . . . 6 ⊢ ((𝐺 ↾s 𝑆) ∈ Grp → (0g‘(𝐺 ↾s 𝑆)) ∈ (Base‘(𝐺 ↾s 𝑆)))
8	4, 7	syl 14	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘(𝐺 ↾s 𝑆)) ∈ (Base‘(𝐺 ↾s 𝑆)))
9	3	subgbas 13828	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆)))
10	8, 9	eleqtrrd 2311	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘(𝐺 ↾s 𝑆)) ∈ 𝑆)
11		elex2 2820	. . . 4 ⊢ ((0g‘(𝐺 ↾s 𝑆)) ∈ 𝑆 → ∃𝑢 𝑢 ∈ 𝑆)
12	10, 11	syl 14	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∃𝑢 𝑢 ∈ 𝑆)
13		issubg2.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
14	13	subgcl 13834	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
15	14	3expa 1230	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
16	15	ralrimiva 2606	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
17		issubg2.i	. . . . . 6 ⊢ 𝐼 = (invg‘𝐺)
18	17	subginvcl 13833	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆)
19	16, 18	jca 306	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) → (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))
20	19	ralrimiva 2606	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))
21	2, 12, 20	3jca 1204	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ⊆ 𝐵 ∧ ∃𝑢 𝑢 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)))
22		eleq1w 2292	. . . . 5 ⊢ (𝑟 = 𝑢 → (𝑟 ∈ 𝑆 ↔ 𝑢 ∈ 𝑆))
23	22	cbvexv 1967	. . . 4 ⊢ (∃𝑟 𝑟 ∈ 𝑆 ↔ ∃𝑢 𝑢 ∈ 𝑆)
24	23	3anbi2i 1218	. . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)) ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑢 𝑢 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)))
25		simpl 109	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → 𝐺 ∈ Grp)
26		simpr1 1030	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → 𝑆 ⊆ 𝐵)
27	3	a1i 9	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆))
28	1	a1i 9	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐵 = (Base‘𝐺))
29		simpl 109	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐺 ∈ Grp)
30		simpr 110	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ 𝐵)
31	27, 28, 29, 30	ressbas2d 13214	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = (Base‘(𝐺 ↾s 𝑆)))
32	31	3ad2antr1 1189	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → 𝑆 = (Base‘(𝐺 ↾s 𝑆)))
33	13	a1i 9	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → + = (+g‘𝐺))
34		basfn 13204	. . . . . . . . . . 11 ⊢ Base Fn V
35	29	elexd 2817	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐺 ∈ V)
36		funfvex 5665	. . . . . . . . . . . 12 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
37	36	funfni 5439	. . . . . . . . . . 11 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
38	34, 35, 37	sylancr 414	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (Base‘𝐺) ∈ V)
39	1, 38	eqeltrid 2318	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝐵 ∈ V)
40	39, 30	ssexd 4234	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ V)
41	27, 33, 40, 29	ressplusgd 13275	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → + = (+g‘(𝐺 ↾s 𝑆)))
42	41	3ad2antr1 1189	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → + = (+g‘(𝐺 ↾s 𝑆)))
43		simpr3 1032	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))
44		simpl 109	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆) → ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
45	44	ralimi 2596	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
46	43, 45	syl 14	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
47		oveq1 6035	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥 + 𝑦) = (𝑢 + 𝑦))
48	47	eleq1d 2300	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑢 + 𝑦) ∈ 𝑆))
49		oveq2 6036	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑢 + 𝑦) = (𝑢 + 𝑣))
50	49	eleq1d 2300	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑢 + 𝑦) ∈ 𝑆 ↔ (𝑢 + 𝑣) ∈ 𝑆))
51	48, 50	rspc2v 2924	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 → (𝑢 + 𝑣) ∈ 𝑆))
52	46, 51	syl5com 29	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 + 𝑣) ∈ 𝑆))
53	52	3impib 1228	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 + 𝑣) ∈ 𝑆)
54	26	sseld 3227	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → (𝑢 ∈ 𝑆 → 𝑢 ∈ 𝐵))
55	26	sseld 3227	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → (𝑣 ∈ 𝑆 → 𝑣 ∈ 𝐵))
56	26	sseld 3227	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → (𝑤 ∈ 𝑆 → 𝑤 ∈ 𝐵))
57	54, 55, 56	3anim123d 1356	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
58	57	imp 124	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
59	1, 13	grpass 13655	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
60	59	adantlr 477	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
61	58, 60	syldan 282	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
62		simpr2 1031	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ∃𝑟 𝑟 ∈ 𝑆)
63	62, 23	sylib 122	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ∃𝑢 𝑢 ∈ 𝑆)
64	26	sselda 3228	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → 𝑢 ∈ 𝐵)
65		eqid 2231	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
66	1, 13, 65, 17	grplinv 13696	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((𝐼‘𝑢) + 𝑢) = (0g‘𝐺))
67	66	adantlr 477	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝐵) → ((𝐼‘𝑢) + 𝑢) = (0g‘𝐺))
68	64, 67	syldan 282	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → ((𝐼‘𝑢) + 𝑢) = (0g‘𝐺))
69		simpr 110	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆)
70	69	ralimi 2596	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆) → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆)
71	43, 70	syl 14	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆)
72		fveq2 5648	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝐼‘𝑥) = (𝐼‘𝑢))
73	72	eleq1d 2300	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → ((𝐼‘𝑥) ∈ 𝑆 ↔ (𝐼‘𝑢) ∈ 𝑆))
74	73	rspccva 2910	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (𝐼‘𝑢) ∈ 𝑆)
75	71, 74	sylan 283	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → (𝐼‘𝑢) ∈ 𝑆)
76		simpr 110	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → 𝑢 ∈ 𝑆)
77	46	adantr 276	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
78		ovrspc2v 6054	. . . . . . . . 9 ⊢ ((((𝐼‘𝑢) ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) → ((𝐼‘𝑢) + 𝑢) ∈ 𝑆)
79	75, 76, 77, 78	syl21anc 1273	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → ((𝐼‘𝑢) + 𝑢) ∈ 𝑆)
80	68, 79	eqeltrrd 2309	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → (0g‘𝐺) ∈ 𝑆)
81	63, 80	exlimddv 1947	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → (0g‘𝐺) ∈ 𝑆)
82	1, 13, 65	grplid 13677	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
83	82	adantlr 477	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
84	64, 83	syldan 282	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) ∧ 𝑢 ∈ 𝑆) → ((0g‘𝐺) + 𝑢) = 𝑢)
85	32, 42, 53, 61, 81, 84, 75, 68	isgrpd 13669	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → (𝐺 ↾s 𝑆) ∈ Grp)
86	1	issubg 13823	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
87	25, 26, 85, 86	syl3anbrc 1208	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺))
88	87	ex 115	. . 3 ⊢ (𝐺 ∈ Grp → ((𝑆 ⊆ 𝐵 ∧ ∃𝑟 𝑟 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺)))
89	24, 88	biimtrrid 153	. 2 ⊢ (𝐺 ∈ Grp → ((𝑆 ⊆ 𝐵 ∧ ∃𝑢 𝑢 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺)))
90	21, 89	impbid2 143	1 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑢 𝑢 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  Basecbs 13145   ↾_s cress 13146  +_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:  issubgrpd2  13840  issubg3  13842  issubg4m  13843  grpissubg  13844  subgintm  13848  nmzsubg  13860  ghmrn  13907  ghmpreima  13916  subrgugrp  14318  lsssubg  14456  lidlsubg  14565  cnsubglem  14658  mplsubgfi  14785