torsubg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > torsubg		Structured version Visualization version GIF version

Theorem torsubg 19770

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 6080	. . . 4 ⊢ (^◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2731	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19453	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ₀
5	4	fdmi 6729	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 4018	. . 3 ⊢ (^◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19701	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2731	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
10	2, 9	grpidcl 18893	. . . . 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 19475	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0_g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) = 1)
14		1nn 12230	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2840	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) ∈ ℕ)
16		ffn 6717	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ₀ → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7059	. . . . 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 582	. . 3 ⊢ (𝐺 ∈ Abel → (0_g‘𝐺) ∈ (^◡𝑂 “ ℕ))
21	20	ne0d 4335	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 723	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3978	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡𝑂 “ ℕ) → 𝑥 ∈ (Base‘𝐺))
24	23	ad2antlr 724	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝑥 ∈ (Base‘𝐺))
25	6	sseli 3978	. . . . . . . 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 2731	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
28	2, 27	grpcl 18869	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
29	22, 24, 26, 28	syl3anc 1370	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
30		0nnn 12255	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19452	. . . . . . . . . . . . . . . . 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 12591	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19452	. . . . . . . . . . . . . . . . 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 12591	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16456	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ₀)
38	37	nn0cnd 12541	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11419	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5158	. . . . . . . . . . 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 12679	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16227	. . . . . . . . . . . 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 7059	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
49		elpreima 7059	. . . . . . . . . . . . . . 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 12272	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2820	. . . . . . . . . . 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 764	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝐺 ∈ Abel)
59	3, 2, 27	odadd1 19764	. . . . . . . . . 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 1370	. . . . . . . . 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 7419	. . . . . . . . . 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 5158	. . . . . . . . 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 19452	. . . . . . . . . 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 12481	. . . . . . . . 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 861	. . . . . . 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 7059	. . . . . . 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 582	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ))
74	73	ralrimiva 3145	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ))
75		eqid 2731	. . . . . . 7 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
76	2, 75	grpinvcl 18915	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺))
77	8, 23, 76	syl2an 595	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺))
78	3, 75, 2	odinv 19477	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) = (𝑂‘𝑥))
79	8, 23, 78	syl2an 595	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) = (𝑂‘𝑥))
80	47	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
81	79, 80	eqeltrd 2832	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7059	. . . . . 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 582	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ))
85	74, 84	jca 511	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ) ∧ ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ)))
86	85	ralrimiva 3145	. 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 19064	. . 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 1341	1 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 · cmul 11121 ℕcn 12219 ℕ₀cn0 12479 ℤcz 12565 ∥ cdvds 16204 gcd cgcd 16442 Basecbs 17151 +_gcplusg 17204 0_gc0g 17392 Grpcgrp 18861 inv_gcminusg 18862 SubGrpcsubg 19043 odcod 19440 Abelcabl 19697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-gcd 16443 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-od 19444 df-cmn 19698 df-abl 19699

This theorem is referenced by: (None)

W3C validator