Theorem torsubg 19833

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 6069	. . . 4 ⊢ (◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2735	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19516	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ0
5	4	fdmi 6716	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 4007	. . 3 ⊢ (◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19764	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2735	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	2, 9	grpidcl 18946	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
11	8, 10	syl 17	. . . 4 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (Base‘𝐺))
12	3, 9	od1 19538	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) = 1)
14		1nn 12249	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2842	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) ∈ ℕ)
16		ffn 6705	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ0 → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7047	. . . . 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 4317	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3954	. . . . . . . 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 3954	. . . . . . . 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 2735	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
28	2, 27	grpcl 18922	. . . . . . 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 12274	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19515	. . . . . . . . . . . . . . . . 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 12612	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19515	. . . . . . . . . . . . . . . . 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 12612	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16525	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ0)
38	37	nn0cnd 12562	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11431	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5129	. . . . . . . . . . 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 12701	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16294	. . . . . . . . . . . 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 7047	. . . . . . . . . . . . . . 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 7047	. . . . . . . . . . . . . . 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 12291	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2822	. . . . . . . . . . 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 19827	. . . . . . . . . 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 7410	. . . . . . . . . 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 5129	. . . . . . . . 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 19515	. . . . . . . . . 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 12501	. . . . . . . . 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 7047	. . . . . . 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 3132	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
75		eqid 2735	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
76	2, 75	grpinvcl 18968	. . . . . 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 19540	. . . . . . 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 2834	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7047	. . . . . 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 3132	. 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 19122	. . 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 2108   ≠ wne 2932  ∀wral 3051   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119  ^◡ccnv 5653  dom cdm 5654   “ cima 5657   Fn wfn 6525  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403  0cc0 11127  1c1 11128   · cmul 11132  ℕcn 12238  ℕ₀cn0 12499  ℤcz 12586   ∥ cdvds 16270   gcd cgcd 16511  Basecbs 17226  +_gcplusg 17269  0_gc0g 17451  Grpcgrp 18914  inv_gcminusg 18915  SubGrpcsubg 19101  odcod 19503  Abelcabl 19760

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9452  df-inf 9453  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-n0 12500  df-z 12587  df-uz 12851  df-rp 13007  df-fz 13523  df-fzo 13670  df-fl 13807  df-mod 13885  df-seq 14018  df-exp 14078  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-dvds 16271  df-gcd 16512  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-0g 17453  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-grp 18917  df-minusg 18918  df-sbg 18919  df-mulg 19049  df-subg 19104  df-od 19507  df-cmn 19761  df-abl 19762

This theorem is referenced by: (None)