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Theorem torsubg 19716

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 6077	. . . 4 ⊢ (^◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2732	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19399	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ₀
5	4	fdmi 6726	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 4017	. . 3 ⊢ (^◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19647	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2732	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
10	2, 9	grpidcl 18846	. . . . 5 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ (Base‘𝐺))
11	8, 10	syl 17	. . . 4 ⊢ (𝐺 ∈ Abel → (0_g‘𝐺) ∈ (Base‘𝐺))
12	3, 9	od1 19421	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0_g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) = 1)
14		1nn 12219	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2841	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) ∈ ℕ)
16		ffn 6714	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ₀ → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7056	. . . . 5 ⊢ (𝑂 Fn (Base‘𝐺) → ((0_g‘𝐺) ∈ (^◡𝑂 “ ℕ) ↔ ((0_g‘𝐺) ∈ (Base‘𝐺) ∧ (𝑂‘(0_g‘𝐺)) ∈ ℕ)))
19	17, 18	ax-mp 5	. . . 4 ⊢ ((0_g‘𝐺) ∈ (^◡𝑂 “ ℕ) ↔ ((0_g‘𝐺) ∈ (Base‘𝐺) ∧ (𝑂‘(0_g‘𝐺)) ∈ ℕ))
20	11, 15, 19	sylanbrc 583	. . 3 ⊢ (𝐺 ∈ Abel → (0_g‘𝐺) ∈ (^◡𝑂 “ ℕ))
21	20	ne0d 4334	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 724	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3977	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
24	23	ad2antlr 725	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
25	6	sseli 3977	. . . . . . . 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 2732	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
28	2, 27	grpcl 18823	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
29	22, 24, 26, 28	syl3anc 1371	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
30		0nnn 12244	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19398	. . . . . . . . . . . . . . . . 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 12580	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19398	. . . . . . . . . . . . . . . . 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 12580	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16445	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ₀)
38	37	nn0cnd 12530	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11408	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5157	. . . . . . . . . . 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 12668	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16216	. . . . . . . . . . . 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 278	. . . . . . . . . 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 7056	. . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (^◡𝑂 “ ℕ) → (𝑂‘𝑥) ∈ ℕ)
48	47	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
49		elpreima 7056	. . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . 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 12261	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2821	. . . . . . . . . . 11 ⊢ (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → (((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ ↔ 0 ∈ ℕ))
55	53, 54	syl5ibcom 244	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (((𝑂‘𝑥) · (𝑂‘𝑦)) = 0 → 0 ∈ ℕ))
56	44, 55	sylbid 239	. . . . . . . . 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 198	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ¬ (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) ∥ ((𝑂‘𝑥) · (𝑂‘𝑦)))
58		simpll 765	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝐺 ∈ Abel)
59	3, 2, 27	odadd1 19710	. . . . . . . . . 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 1371	. . . . . . . . 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 7412	. . . . . . . . . 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 5157	. . . . . . . . 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 244	. . . . . . . 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 197	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ¬ (𝑂‘(𝑥(+_g‘𝐺)𝑦)) = 0)
65	2, 3	odcl 19398	. . . . . . . . . 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 12470	. . . . . . . . 9 ⊢ ((𝑂‘(𝑥(+_g‘𝐺)𝑦)) ∈ ℕ₀ ↔ ((𝑂‘(𝑥(+_g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+_g‘𝐺)𝑦)) = 0))
68	66, 67	sylib 217	. . . . . . . 8 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘(𝑥(+_g‘𝐺)𝑦)) ∈ ℕ ∨ (𝑂‘(𝑥(+_g‘𝐺)𝑦)) = 0))
69	68	ord 862	. . . . . . 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 7056	. . . . . . 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 3146	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ))
75		eqid 2732	. . . . . . 7 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
76	2, 75	grpinvcl 18868	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺))
77	8, 23, 76	syl2an 596	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺))
78	3, 75, 2	odinv 19423	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) = (𝑂‘𝑥))
79	8, 23, 78	syl2an 596	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) = (𝑂‘𝑥))
80	47	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
81	79, 80	eqeltrd 2833	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7056	. . . . . 6 ⊢ (𝑂 Fn (Base‘𝐺) → (((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ) ↔ (((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((inv_g‘𝐺)‘𝑥)) ∈ ℕ)))
83	17, 82	ax-mp 5	. . . . 5 ⊢ (((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ) ↔ (((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺) ∧ (𝑂‘((inv_g‘𝐺)‘𝑥)) ∈ ℕ))
84	77, 81, 83	sylanbrc 583	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ))
85	74, 84	jca 512	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ) ∧ ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ)))
86	85	ralrimiva 3146	. 2 ⊢ (𝐺 ∈ Abel → ∀𝑥 ∈ (^◡𝑂 “ ℕ)(∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ) ∧ ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ)))
87	2, 27, 75	issubg2 19015	. . 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) ∧ ((inv_g‘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) ∧ ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ)))))
89	7, 21, 86, 88	mpbir3and 1342	1 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 ∅c0 4321 class class class wbr 5147 ^◡ccnv 5674 dom cdm 5675 “ cima 5678 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 · cmul 11111 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ∥ cdvds 16193 gcd cgcd 16431 Basecbs 17140 +_gcplusg 17193 0_gc0g 17381 Grpcgrp 18815 inv_gcminusg 18816 SubGrpcsubg 18994 odcod 19386 Abelcabl 19643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-od 19390 df-cmn 19644 df-abl 19645

This theorem is referenced by: (None)

W3C validator