Theorem torsubg 19760

Description: The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)

Hypothesis

Ref	Expression
torsubg.1	⊢ 𝑂 = (od‘𝐺)

Assertion

Ref	Expression
torsubg	⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Proof of Theorem torsubg

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

Step	Hyp	Ref	Expression
1		cnvimass 6042	. . . 4 ⊢ (◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2729	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19443	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ0
5	4	fdmi 6681	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 3992	. . 3 ⊢ (◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19691	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2729	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	2, 9	grpidcl 18873	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
11	8, 10	syl 17	. . . 4 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (Base‘𝐺))
12	3, 9	od1 19465	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) = 1)
14		1nn 12173	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2836	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) ∈ ℕ)
16		ffn 6670	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ0 → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7012	. . . . 5 ⊢ (𝑂 Fn (Base‘𝐺) → ((0g‘𝐺) ∈ (◡𝑂 “ ℕ) ↔ ((0g‘𝐺) ∈ (Base‘𝐺) ∧ (𝑂‘(0g‘𝐺)) ∈ ℕ)))
19	17, 18	ax-mp 5	. . . 4 ⊢ ((0g‘𝐺) ∈ (◡𝑂 “ ℕ) ↔ ((0g‘𝐺) ∈ (Base‘𝐺) ∧ (𝑂‘(0g‘𝐺)) ∈ ℕ))
20	11, 15, 19	sylanbrc 583	. . 3 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (◡𝑂 “ ℕ))
21	20	ne0d 4301	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3939	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
24	23	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
25	6	sseli 3939	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
26	25	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27		eqid 2729	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
28	2, 27	grpcl 18849	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
29	22, 24, 26, 28	syl3anc 1373	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
30		0nnn 12198	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19442	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (Base‘𝐺) → (𝑂‘𝑥) ∈ ℕ0)
32	24, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ0)
33	32	nn0zd 12531	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19442	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (Base‘𝐺) → (𝑂‘𝑦) ∈ ℕ0)
35	26, 34	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℕ0)
36	35	nn0zd 12531	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16454	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ0)
38	37	nn0cnd 12481	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11348	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5112	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ 0 ∥ ((𝑂‘𝑥) · (𝑂‘𝑦))))
41	33, 36	zmulcld 12620	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16222	. . . . . . . . . . . 12 ⊢ (((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ → (0 ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ ((𝑂‘𝑥) · (𝑂‘𝑦)) = 0))
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (0 ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ ((𝑂‘𝑥) · (𝑂‘𝑦)) = 0))
44	40, 43	bitrd 279	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ ((𝑂‘𝑥) · (𝑂‘𝑦)) = 0))
45		elpreima 7012	. . . . . . . . . . . . . . 15 ⊢ (𝑂 Fn (Base‘𝐺) → (𝑥 ∈ (◡𝑂 “ ℕ) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝑂‘𝑥) ∈ ℕ)))
46	17, 45	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝑂‘𝑥) ∈ ℕ))
47	46	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) → (𝑂‘𝑥) ∈ ℕ)
48	47	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
49		elpreima 7012	. . . . . . . . . . . . . . 15 ⊢ (𝑂 Fn (Base‘𝐺) → (𝑦 ∈ (◡𝑂 “ ℕ) ↔ (𝑦 ∈ (Base‘𝐺) ∧ (𝑂‘𝑦) ∈ ℕ)))
50	17, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) ↔ (𝑦 ∈ (Base‘𝐺) ∧ (𝑂‘𝑦) ∈ ℕ))
51	50	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) → (𝑂‘𝑦) ∈ ℕ)
52	51	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℕ)
53	48, 52	nnmulcld 12215	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2816	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → (((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
55	53, 54	syl5ibcom 245	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → 0 ∈ ℕ))
56	44, 55	sylbid 240	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) → 0 ∈ ℕ))
57	30, 56	mtoi 199	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ¬ (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)))
58		simpll 766	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Abel)
59	3, 2, 27	odadd1 19754	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)))
60	58, 24, 26, 59	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)))
61		oveq1 7376	. . . . . . . . . 10 ⊢ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0 → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))))
62	61	breq1d 5112	. . . . . . . . 9 ⊢ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0 → (((𝑂‘(𝑥(+g‘𝐺)𝑦)) · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦))))
63	60, 62	syl5ibcom 245	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0 → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦))))
64	57, 63	mtod 198	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0)
65	2, 3	odcl 19442	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) → (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0)
66	29, 65	syl 17	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0)
67		elnn0 12420	. . . . . . . . 9 ⊢ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
68	66, 67	sylib 218	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
69	68	ord 864	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (¬ (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ → (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
70	64, 69	mt3d 148	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ)
71		elpreima 7012	. . . . . . 7 ⊢ (𝑂 Fn (Base‘𝐺) → ((𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ↔ ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ)))
72	17, 71	ax-mp 5	. . . . . 6 ⊢ ((𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ↔ ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ))
73	29, 70, 72	sylanbrc 583	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
74	73	ralrimiva 3125	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
75		eqid 2729	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
76	2, 75	grpinvcl 18895	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
77	8, 23, 76	syl2an 596	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
78	3, 75, 2	odinv 19467	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg‘𝐺)‘𝑥)) = (𝑂‘𝑥))
79	8, 23, 78	syl2an 596	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) = (𝑂‘𝑥))
80	47	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
81	79, 80	eqeltrd 2828	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7012	. . . . . 6 ⊢ (𝑂 Fn (Base‘𝐺) → (((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ) ↔ (((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((invg‘𝐺)‘𝑥)) ∈ ℕ)))
83	17, 82	ax-mp 5	. . . . 5 ⊢ (((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ) ↔ (((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((invg‘𝐺)‘𝑥)) ∈ ℕ))
84	77, 81, 83	sylanbrc 583	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ))
85	74, 84	jca 511	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ∧ ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ)))
86	85	ralrimiva 3125	. 2 ⊢ (𝐺 ∈ Abel → ∀𝑥 ∈ (◡𝑂 “ ℕ)(∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ∧ ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ)))
87	2, 27, 75	issubg2 19049	. . 3 ⊢ (𝐺 ∈ Grp → ((◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺) ↔ ((◡𝑂 “ ℕ) ⊆ (Base‘𝐺) ∧ (◡𝑂 “ ℕ) ≠ ∅ ∧ ∀𝑥 ∈ (◡𝑂 “ ℕ)(∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ∧ ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ)))))
88	8, 87	syl 17	. 2 ⊢ (𝐺 ∈ Abel → ((◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺) ↔ ((◡𝑂 “ ℕ) ⊆ (Base‘𝐺) ∧ (◡𝑂 “ ℕ) ≠ ∅ ∧ ∀𝑥 ∈ (◡𝑂 “ ℕ)(∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ∧ ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ)))))
89	7, 21, 86, 88	mpbir3and 1343	1 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  0cc0 11044  1c1 11045   · cmul 11049  ℕcn 12162  ℕ₀cn0 12418  ℤcz 12505   ∥ cdvds 16198   gcd cgcd 16440  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Grpcgrp 18841  inv_gcminusg 18842  SubGrpcsubg 19028  odcod 19430  Abelcabl 19687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9369  df-inf 9370  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-fz 13445  df-fzo 13592  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-dvds 16199  df-gcd 16441  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-od 19434  df-cmn 19688  df-abl 19689

This theorem is referenced by: (None)