Theorem cycsubgcl 19049

Description: The set of integer powers of an element 𝐴 of a group forms a subgroup containing 𝐴, called the cyclic group generated by the element 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
cycsubg.x	⊢ 𝑋 = (Base‘𝐺)
cycsubg.t	⊢ · = (.g‘𝐺)
cycsubg.f	⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))

Assertion

Ref	Expression
cycsubgcl	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥, ·   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem cycsubgcl

Dummy variables 𝑚 𝑛 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycsubg.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
2		cycsubg.t	. . . . . . . 8 ⊢ · = (.g‘𝐺)
3	1, 2	mulgcl 18943	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
4	3	3expa 1118	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ ℤ) ∧ 𝐴 ∈ 𝑋) → (𝑥 · 𝐴) ∈ 𝑋)
5	4	an32s 650	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
6		cycsubg.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
7	5, 6	fmptd 7098	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:ℤ⟶𝑋)
8	7	frnd 6712	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ⊆ 𝑋)
9	7	ffnd 6705	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn ℤ)
10		1z 12574	. . . . 5 ⊢ 1 ∈ ℤ
11		fnfvelrn 7067	. . . . 5 ⊢ ((𝐹 Fn ℤ ∧ 1 ∈ ℤ) → (𝐹‘1) ∈ ran 𝐹)
12	9, 10, 11	sylancl 586	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘1) ∈ ran 𝐹)
13	12	ne0d 4331	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ≠ ∅)
14		df-3an 1089	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴 ∈ 𝑋) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴 ∈ 𝑋))
15		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	1, 2, 15	mulgdir 18958	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴 ∈ 𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g‘𝐺)(𝑛 · 𝐴)))
17	14, 16	sylan2br 595	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴 ∈ 𝑋)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g‘𝐺)(𝑛 · 𝐴)))
18	17	anass1rs 653	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g‘𝐺)(𝑛 · 𝐴)))
19		zaddcl 12584	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + 𝑛) ∈ ℤ)
20	19	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝑚 + 𝑛) ∈ ℤ)
21		oveq1 7400	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑚 + 𝑛) → (𝑥 · 𝐴) = ((𝑚 + 𝑛) · 𝐴))
22		ovex 7426	. . . . . . . . . . . . 13 ⊢ ((𝑚 + 𝑛) · 𝐴) ∈ V
23	21, 6, 22	fvmpt 6984	. . . . . . . . . . . 12 ⊢ ((𝑚 + 𝑛) ∈ ℤ → (𝐹‘(𝑚 + 𝑛)) = ((𝑚 + 𝑛) · 𝐴))
24	20, 23	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝑚 + 𝑛) · 𝐴))
25		oveq1 7400	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑚 → (𝑥 · 𝐴) = (𝑚 · 𝐴))
26		ovex 7426	. . . . . . . . . . . . . 14 ⊢ (𝑚 · 𝐴) ∈ V
27	25, 6, 26	fvmpt 6984	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℤ → (𝐹‘𝑚) = (𝑚 · 𝐴))
28	27	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘𝑚) = (𝑚 · 𝐴))
29		oveq1 7400	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (𝑥 · 𝐴) = (𝑛 · 𝐴))
30		ovex 7426	. . . . . . . . . . . . . 14 ⊢ (𝑛 · 𝐴) ∈ V
31	29, 6, 30	fvmpt 6984	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝐹‘𝑛) = (𝑛 · 𝐴))
32	31	ad2antll 727	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘𝑛) = (𝑛 · 𝐴))
33	28, 32	oveq12d 7411	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) = ((𝑚 · 𝐴)(+g‘𝐺)(𝑛 · 𝐴)))
34	18, 24, 33	3eqtr4d 2781	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)))
35		fnfvelrn 7067	. . . . . . . . . . 11 ⊢ ((𝐹 Fn ℤ ∧ (𝑚 + 𝑛) ∈ ℤ) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
36	9, 19, 35	syl2an 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) ∈ ran 𝐹)
37	34, 36	eqeltrrd 2833	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹)
38	37	anassrs 468	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹)
39	38	ralrimiva 3145	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑛 ∈ ℤ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹)
40		oveq2 7401	. . . . . . . . . . 11 ⊢ (𝑣 = (𝐹‘𝑛) → ((𝐹‘𝑚)(+g‘𝐺)𝑣) = ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)))
41	40	eleq1d 2817	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑛) → (((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹))
42	41	ralrn 7074	. . . . . . . . 9 ⊢ (𝐹 Fn ℤ → (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹))
43	9, 42	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹))
44	43	adantr 481	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑛 ∈ ℤ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹))
45	39, 44	mpbird 256	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹)
46		eqid 2731	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
47	1, 2, 46	mulgneg 18944	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → (-𝑚 · 𝐴) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
48	47	3expa 1118	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ∈ 𝑋) → (-𝑚 · 𝐴) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
49	48	an32s 650	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (-𝑚 · 𝐴) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
50		znegcl 12579	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
51	50	adantl 482	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → -𝑚 ∈ ℤ)
52		oveq1 7400	. . . . . . . . . 10 ⊢ (𝑥 = -𝑚 → (𝑥 · 𝐴) = (-𝑚 · 𝐴))
53		ovex 7426	. . . . . . . . . 10 ⊢ (-𝑚 · 𝐴) ∈ V
54	52, 6, 53	fvmpt 6984	. . . . . . . . 9 ⊢ (-𝑚 ∈ ℤ → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
55	51, 54	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = (-𝑚 · 𝐴))
56	27	adantl 482	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) = (𝑚 · 𝐴))
57	56	fveq2d 6882	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ((invg‘𝐺)‘(𝐹‘𝑚)) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
58	49, 55, 57	3eqtr4d 2781	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg‘𝐺)‘(𝐹‘𝑚)))
59		fnfvelrn 7067	. . . . . . . 8 ⊢ ((𝐹 Fn ℤ ∧ -𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
60	9, 50, 59	syl2an 596	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) ∈ ran 𝐹)
61	58, 60	eqeltrrd 2833	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹)
62	45, 61	jca 512	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹))
63	62	ralrimiva 3145	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹))
64		oveq1 7400	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑚) → (𝑢(+g‘𝐺)𝑣) = ((𝐹‘𝑚)(+g‘𝐺)𝑣))
65	64	eleq1d 2817	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑚) → ((𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹))
66	65	ralbidv 3176	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑚) → (∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹))
67		fveq2 6878	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑚) → ((invg‘𝐺)‘𝑢) = ((invg‘𝐺)‘(𝐹‘𝑚)))
68	67	eleq1d 2817	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑚) → (((invg‘𝐺)‘𝑢) ∈ ran 𝐹 ↔ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹))
69	66, 68	anbi12d 631	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑚) → ((∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘𝑢) ∈ ran 𝐹) ↔ (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹)))
70	69	ralrn 7074	. . . . 5 ⊢ (𝐹 Fn ℤ → (∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘𝑢) ∈ ran 𝐹) ↔ ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹)))
71	9, 70	syl 17	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘𝑢) ∈ ran 𝐹) ↔ ∀𝑚 ∈ ℤ (∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹)))
72	63, 71	mpbird 256	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘𝑢) ∈ ran 𝐹))
73	1, 15, 46	issubg2 18993	. . . 4 ⊢ (𝐺 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝐺) ↔ (ran 𝐹 ⊆ 𝑋 ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘𝑢) ∈ ran 𝐹))))
74	73	adantr 481	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ↔ (ran 𝐹 ⊆ 𝑋 ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑢 ∈ ran 𝐹(∀𝑣 ∈ ran 𝐹(𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ∧ ((invg‘𝐺)‘𝑢) ∈ ran 𝐹))))
75	8, 13, 72, 74	mpbir3and 1342	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))
76		oveq1 7400	. . . . . 6 ⊢ (𝑥 = 1 → (𝑥 · 𝐴) = (1 · 𝐴))
77		ovex 7426	. . . . . 6 ⊢ (1 · 𝐴) ∈ V
78	76, 6, 77	fvmpt 6984	. . . . 5 ⊢ (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
79	10, 78	ax-mp 5	. . . 4 ⊢ (𝐹‘1) = (1 · 𝐴)
80	1, 2	mulg1 18933	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (1 · 𝐴) = 𝐴)
81	80	adantl 482	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (1 · 𝐴) = 𝐴)
82	79, 81	eqtrid 2783	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘1) = 𝐴)
83	82, 12	eqeltrrd 2833	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ran 𝐹)
84	75, 83	jca 512	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060   ⊆ wss 3944  ∅c0 4318   ↦ cmpt 5224  ran crn 5670   Fn wfn 6527  ‘cfv 6532  (class class class)co 7393  1c1 11093   + caddc 11095  -cneg 11427  ℤcz 12540  Basecbs 17126  +_gcplusg 17179  Grpcgrp 18794  inv_gcminusg 18795  ._gcmg 18922  SubGrpcsubg 18972

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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

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 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 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-seq 13949  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-minusg 18798  df-mulg 18923  df-subg 18975

This theorem is referenced by:  cycsubg  19051  cycsubgcld  19052  oddvds2  19398  cycsubgcyg  19728