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Theorem torsubg 19774

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 6074	. . . 4 ⊢ (^◡𝑂 “ ℕ) ⊆ dom 𝑂
2		eqid 2726	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		torsubg.1	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odf 19457	. . . . 5 ⊢ 𝑂:(Base‘𝐺)⟶ℕ₀
5	4	fdmi 6723	. . . 4 ⊢ dom 𝑂 = (Base‘𝐺)
6	1, 5	sseqtri 4013	. . 3 ⊢ (^◡𝑂 “ ℕ) ⊆ (Base‘𝐺)
7	6	a1i 11	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ⊆ (Base‘𝐺))
8		ablgrp 19705	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9		eqid 2726	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
10	2, 9	grpidcl 18895	. . . . 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 19479	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑂‘(0_g‘𝐺)) = 1)
13	8, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) = 1)
14		1nn 12227	. . . . 5 ⊢ 1 ∈ ℕ
15	13, 14	eqeltrdi 2835	. . . 4 ⊢ (𝐺 ∈ Abel → (𝑂‘(0_g‘𝐺)) ∈ ℕ)
16		ffn 6711	. . . . . 6 ⊢ (𝑂:(Base‘𝐺)⟶ℕ₀ → 𝑂 Fn (Base‘𝐺))
17	4, 16	ax-mp 5	. . . . 5 ⊢ 𝑂 Fn (Base‘𝐺)
18		elpreima 7053	. . . . 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 4330	. 2 ⊢ (𝐺 ∈ Abel → (^◡𝑂 “ ℕ) ≠ ∅)
22	8	ad2antrr 723	. . . . . . 7 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → 𝐺 ∈ Grp)
23	6	sseli 3973	. . . . . . . 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 3973	. . . . . . . 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 2726	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
28	2, 27	grpcl 18871	. . . . . . 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 12252	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℕ
31	2, 3	odcl 19456	. . . . . . . . . . . . . . . . 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 12588	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘𝑥) ∈ ℤ)
34	2, 3	odcl 19456	. . . . . . . . . . . . . . . . 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 12588	. . . . . . . . . . . . . . 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 12538	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) gcd (𝑂‘𝑦)) ∈ ℂ)
39	38	mul02d 11416	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → (0 · ((𝑂‘𝑥) gcd (𝑂‘𝑦))) = 0)
40	39	breq1d 5151	. . . . . . . . . . 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 12676	. . . . . . . . . . . 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 7053	. . . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . . . 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 12269	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) ∧ 𝑦 ∈ (^◡𝑂 “ ℕ)) → ((𝑂‘𝑥) · (𝑂‘𝑦)) ∈ ℕ)
54		eleq1 2815	. . . . . . . . . . 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 19768	. . . . . . . . . 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 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 5151	. . . . . . . . 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 19456	. . . . . . . . . 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 12478	. . . . . . . . 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 7053	. . . . . . 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 3140	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → ∀𝑦 ∈ (^◡𝑂 “ ℕ)(𝑥(+_g‘𝐺)𝑦) ∈ (^◡𝑂 “ ℕ))
75		eqid 2726	. . . . . . 7 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
76	2, 75	grpinvcl 18917	. . . . . 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 19481	. . . . . . 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 2827	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ (^◡𝑂 “ ℕ)) → (𝑂‘((inv_g‘𝐺)‘𝑥)) ∈ ℕ)
82		elpreima 7053	. . . . . 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 3140	. 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 19068	. . 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 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 ^◡ccnv 5668 dom cdm 5669 “ cima 5672 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 · cmul 11117 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ∥ cdvds 16204 gcd cgcd 16442 Basecbs 17153 +_gcplusg 17206 0_gc0g 17394 Grpcgrp 18863 inv_gcminusg 18864 SubGrpcsubg 19047 odcod 19444 Abelcabl 19701

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-gcd 16443 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-od 19448 df-cmn 19702 df-abl 19703

This theorem is referenced by: (None)

W3C validator