Theorem torsubg 19820

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 6034	. . . 4 ⊢ (◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2739	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19503	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ0
5	4	fdmi 6666	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 3963	. . 3 ⊢ (◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19751	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2739	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	2, 9	grpidcl 18932	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
11	8, 10	syl 17	. . . 4 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (Base‘𝐺))
12	3, 9	od1 19525	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) = 1)
14		1nn 12176	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2847	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) ∈ ℕ)
16		ffn 6655	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ0 → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 6999	. . . . 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 589	. . 3 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (◡𝑂 “ ℕ))
21	20	ne0d 4270	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 732	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3911	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
24	23	ad2antlr 733	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
25	6	sseli 3911	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
26	25	adantl 482	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27		eqid 2739	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
28	2, 27	grpcl 18908	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
29	22, 24, 26, 28	syl3anc 1379	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
30		0nnn 12204	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19502	. . . . . . . . . . . . . . . . 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 12540	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19502	. . . . . . . . . . . . . . . . 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 12540	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16468	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ0)
38	37	nn0cnd 12491	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11335	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5082	. . . . . . . . . . 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 12630	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16236	. . . . . . . . . . . 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 280	. . . . . . . . . 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 6999	. . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) → (𝑂‘𝑥) ∈ ℕ)
48	47	ad2antlr 733	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
49		elpreima 6999	. . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) → (𝑂‘𝑦) ∈ ℕ)
52	51	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℕ)
53	48, 52	nnmulcld 12221	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2827	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → (((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
55	53, 54	syl5ibcom 246	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → 0 ∈ ℕ))
56	44, 55	sylbid 241	. . . . . . . . 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 200	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ¬ (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)))
58		simpll 772	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Abel)
59	3, 2, 27	odadd1 19814	. . . . . . . . . 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 1379	. . . . . . . . 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 7363	. . . . . . . . . 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 5082	. . . . . . . . 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 246	. . . . . . . 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 199	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0)
65	2, 3	odcl 19502	. . . . . . . . . 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 12430	. . . . . . . . 9 ⊢ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
68	66, 67	sylib 219	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
69	68	ord 870	. . . . . . 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 6999	. . . . . . 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 589	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
74	73	ralrimiva 3131	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
75		eqid 2739	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
76	2, 75	grpinvcl 18954	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
77	8, 23, 76	syl2an 602	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
78	3, 75, 2	odinv 19527	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg‘𝐺)‘𝑥)) = (𝑂‘𝑥))
79	8, 23, 78	syl2an 602	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) = (𝑂‘𝑥))
80	47	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
81	79, 80	eqeltrd 2839	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 6999	. . . . . 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 589	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ))
85	74, 84	jca 516	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ∧ ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ)))
86	85	ralrimiva 3131	. 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 19108	. . 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 1349	1 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053   ⊆ wss 3883  ∅c0 4261   class class class wbr 5072  ^◡ccnv 5617  dom cdm 5618   “ cima 5621   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  0cc0 11029  1c1 11030   · cmul 11034  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515   ∥ cdvds 16212   gcd cgcd 16454  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393  Grpcgrp 18900  inv_gcminusg 18901  SubGrpcsubg 19087  odcod 19490  Abelcabl 19747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-gcd 16455  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-od 19494  df-cmn 19748  df-abl 19749

This theorem is referenced by: (None)