Theorem torsubg 20068

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 6198	. . . 4 ⊢ (◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2761	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19751	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ0
5	4	fdmi 6721	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 3979	. . 3 ⊢ (◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19999	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2761	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	2, 9	grpidcl 19176	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
11	8, 10	syl 18	. . . 4 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (Base‘𝐺))
12	3, 9	od1 19773	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0g‘𝐺)) = 1)
13	8, 12	syl 18	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) = 1)
14		1nn 12346	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2869	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0g‘𝐺)) ∈ ℕ)
16		ffn 6709	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ0 → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7057	. . . . 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 595	. . 3 ⊢ (𝐺 ∈ Abel → (0g‘𝐺) ∈ (◡𝑂 “ ℕ))
21	20	ne0d 4288	. 2 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 739	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3927	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
24	23	ad2antlr 740	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
25	6	sseli 3927	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) → 𝑦 ∈ (Base‘𝐺))
26	25	adantl 487	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27		eqid 2761	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
28	2, 27	grpcl 19152	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
29	22, 24, 26, 28	syl3anc 1398	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
30		0nnn 12374	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (Base‘𝐺) → (𝑂‘𝑥) ∈ ℕ0)
32	24, 31	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ0)
33	32	nn0zd 12718	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (Base‘𝐺) → (𝑂‘𝑦) ∈ ℕ0)
35	26, 34	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℕ0)
36	35	nn0zd 12718	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16678	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ0)
38	37	nn0cnd 12669	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11508	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5113	. . . . . . . . . . 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 12809	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16446	. . . . . . . . . . . 12 ⊢ (((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ → (0 ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ ((𝑂‘𝑥) · (𝑂‘𝑦)) = 0))
43	41, 42	syl 18	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (0 ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)) ↔ ((𝑂‘𝑥) · (𝑂‘𝑦)) = 0))
44	40, 43	bitrd 282	. . . . . . . . . 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 7057	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (◡𝑂 “ ℕ) → (𝑂‘𝑥) ∈ ℕ)
48	47	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
49		elpreima 7057	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (◡𝑂 “ ℕ) → (𝑂‘𝑦) ∈ ℕ)
52	51	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℕ)
53	48, 52	nnmulcld 12391	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2849	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → (((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
55	53, 54	syl5ibcom 248	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → 0 ∈ ℕ))
56	44, 55	sylbid 243	. . . . . . . . 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 202	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ¬ (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)))
58		simpll 779	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → 𝐺 ∈ Abel)
59	3, 2, 27	odadd1 20062	. . . . . . . . . 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 1398	. . . . . . . . 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 7427	. . . . . . . . . 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 5113	. . . . . . . . 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 248	. . . . . . . 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 201	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0)
65	2, 3	odcl 19750	. . . . . . . . . 10 ⊢ ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) → (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0)
66	29, 65	syl 18	. . . . . . . . 9 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0)
67		elnn0 12608	. . . . . . . . 9 ⊢ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ0 ↔ ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
68	66, 67	sylib 221	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → ((𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+g‘𝐺)𝑦)) = 0))
69	68	ord 878	. . . . . . 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 149	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑂‘(𝑥(+g‘𝐺)𝑦)) ∈ ℕ)
71		elpreima 7057	. . . . . . 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 595	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (◡𝑂 “ ℕ)) → (𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
74	73	ralrimiva 3155	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ))
75		eqid 2761	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
76	2, 75	grpinvcl 19198	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
77	8, 23, 76	syl2an 608	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ((invg‘𝐺)‘𝑥) ∈ (Base‘𝐺))
78	3, 75, 2	odinv 19775	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((invg‘𝐺)‘𝑥)) = (𝑂‘𝑥))
79	8, 23, 78	syl2an 608	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) = (𝑂‘𝑥))
80	47	adantl 487	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
81	79, 80	eqeltrd 2861	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (𝑂‘((invg‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7057	. . . . . 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 595	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ))
85	74, 84	jca 521	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (◡𝑂 “ ℕ)) → (∀𝑦 ∈ (◡𝑂 “ ℕ)(𝑥(+g‘𝐺)𝑦) ∈ (◡𝑂 “ ℕ) ∧ ((invg‘𝐺)‘𝑥) ∈ (◡𝑂 “ ℕ)))
86	85	ralrimiva 3155	. 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 19352	. . 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 18	. 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 1361	1 ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  ^◡ccnv 5650  dom cdm 5651   “ cima 5654   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  0cc0 11200  1c1 11201   · cmul 11205  ℕcn 12335  ℕ₀cn0 12606  ℤcz 12693   ∥ cdvds 16422   gcd cgcd 16664  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  Grpcgrp 19144  inv_gcminusg 19145  SubGrpcsubg 19330  odcod 19738  Abelcabl 19995

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-dvds 16423  df-gcd 16665  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-od 19742  df-cmn 19996  df-abl 19997

This theorem is used by: (None)