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Theorem torsubg 19816

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 6090	. . . 4 ⊢ (^◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2728	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19499	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ₀
5	4	fdmi 6739	. . . 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 19747	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2728	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
10	2, 9	grpidcl 18929	. . . . 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 19521	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0_g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) = 1)
14		1nn 12261	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2837	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) ∈ ℕ)
16		ffn 6727	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ₀ → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7072	. . . . 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 581	. . 3 ⊢ (𝐺 ∈ Abel → (0_g‘𝐺) ∈ (^◡𝑂 “ ℕ))
21	20	ne0d 4339	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 724	. . . . . . 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 725	. . . . . . 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 480	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝑦 ∈ (Base‘𝐺))
27		eqid 2728	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
28	2, 27	grpcl 18905	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
29	22, 24, 26, 28	syl3anc 1368	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
30		0nnn 12286	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19498	. . . . . . . . . . . . . . . . 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 12622	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19498	. . . . . . . . . . . . . . . . 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 12622	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℤ)
37	33, 36	gcdcld 16490	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℕ₀)
38	37	nn0cnd 12572	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11450	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5162	. . . . . . . . . . 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 12710	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℤ)
42		0dvds 16261	. . . . . . . . . . . 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 7072	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 7072	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (^◡𝑂 “ ℕ) → (𝑂‘𝑦) ∈ ℕ)
52	51	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑦) ∈ ℕ)
53	48, 52	nnmulcld 12303	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2817	. . . . . . . . . . 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 19810	. . . . . . . . . 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 1368	. . . . . . . . 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 7433	. . . . . . . . . 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 5162	. . . . . . . . 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 19498	. . . . . . . . . 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 12512	. . . . . . . . 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 7072	. . . . . . 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 581	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ))
74	73	ralrimiva 3143	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ))
75		eqid 2728	. . . . . . 7 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
76	2, 75	grpinvcl 18951	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺))
77	8, 23, 76	syl2an 594	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ((inv_g‘𝐺)‘𝑥) ∈ (Base‘𝐺))
78	3, 75, 2	odinv 19523	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) = (𝑂‘𝑥))
79	8, 23, 78	syl2an 594	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) = (𝑂‘𝑥))
80	47	adantl 480	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℕ)
81	79, 80	eqeltrd 2829	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7072	. . . . . 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 581	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ))
85	74, 84	jca 510	. . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ) ∧ ((inv_g‘𝐺)‘𝑥) ∈ (^◡𝑂 “ ℕ)))
86	85	ralrimiva 3143	. 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 19103	. . 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 1339	1 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 ^◡ccnv 5681 dom cdm 5682 “ cima 5685 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 · cmul 11151 ℕcn 12250 ℕ₀cn0 12510 ℤcz 12596 ∥ cdvds 16238 gcd cgcd 16476 Basecbs 17187 +_gcplusg 17240 0_gc0g 17428 Grpcgrp 18897 inv_gcminusg 18898 SubGrpcsubg 19082 odcod 19486 Abelcabl 19743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-gcd 16477 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-od 19490 df-cmn 19744 df-abl 19745

This theorem is referenced by: (None)

W3C validator