Theorem cycsubgcl 19077

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 18965	. . . . . . 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 7110	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:ℤ⟶𝑋)
8	7	frnd 6722	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ⊆ 𝑋)
9	7	ffnd 6715	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn ℤ)
10		1z 12588	. . . . 5 ⊢ 1 ∈ ℤ
11		fnfvelrn 7079	. . . . 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 4334	. . 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 2732	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	1, 2, 15	mulgdir 18980	. . . . . . . . . . . . 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 12598	. . . . . . . . . . . . 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 7412	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑚 + 𝑛) → (𝑥 · 𝐴) = ((𝑚 + 𝑛) · 𝐴))
22		ovex 7438	. . . . . . . . . . . . 13 ⊢ ((𝑚 + 𝑛) · 𝐴) ∈ V
23	21, 6, 22	fvmpt 6995	. . . . . . . . . . . 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 7412	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑚 → (𝑥 · 𝐴) = (𝑚 · 𝐴))
26		ovex 7438	. . . . . . . . . . . . . 14 ⊢ (𝑚 · 𝐴) ∈ V
27	25, 6, 26	fvmpt 6995	. . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (𝑥 · 𝐴) = (𝑛 · 𝐴))
30		ovex 7438	. . . . . . . . . . . . . 14 ⊢ (𝑛 · 𝐴) ∈ V
31	29, 6, 30	fvmpt 6995	. . . . . . . . . . . . 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 7423	. . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)))
35		fnfvelrn 7079	. . . . . . . . . . 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 2834	. . . . . . . . 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 3146	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑛 ∈ ℤ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹)
40		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑣 = (𝐹‘𝑛) → ((𝐹‘𝑚)(+g‘𝐺)𝑣) = ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)))
41	40	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑛) → (((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹))
42	41	ralrn 7086	. . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
47	1, 2, 46	mulgneg 18966	. . . . . . . . . 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 12593	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
51	50	adantl 482	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → -𝑚 ∈ ℤ)
52		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑥 = -𝑚 → (𝑥 · 𝐴) = (-𝑚 · 𝐴))
53		ovex 7438	. . . . . . . . . 10 ⊢ (-𝑚 · 𝐴) ∈ V
54	52, 6, 53	fvmpt 6995	. . . . . . . . 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 6892	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ((invg‘𝐺)‘(𝐹‘𝑚)) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
58	49, 55, 57	3eqtr4d 2782	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg‘𝐺)‘(𝐹‘𝑚)))
59		fnfvelrn 7079	. . . . . . . 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 2834	. . . . . 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 3146	. . . 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 7412	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑚) → (𝑢(+g‘𝐺)𝑣) = ((𝐹‘𝑚)(+g‘𝐺)𝑣))
65	64	eleq1d 2818	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑚) → ((𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹))
66	65	ralbidv 3177	. . . . . . 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 6888	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑚) → ((invg‘𝐺)‘𝑢) = ((invg‘𝐺)‘(𝐹‘𝑚)))
68	67	eleq1d 2818	. . . . . . 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 7086	. . . . 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 19015	. . . 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 7412	. . . . . 6 ⊢ (𝑥 = 1 → (𝑥 · 𝐴) = (1 · 𝐴))
77		ovex 7438	. . . . . 6 ⊢ (1 · 𝐴) ∈ V
78	76, 6, 77	fvmpt 6995	. . . . 5 ⊢ (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
79	10, 78	ax-mp 5	. . . 4 ⊢ (𝐹‘1) = (1 · 𝐴)
80	1, 2	mulg1 18955	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (1 · 𝐴) = 𝐴)
81	80	adantl 482	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (1 · 𝐴) = 𝐴)
82	79, 81	eqtrid 2784	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘1) = 𝐴)
83	82, 12	eqeltrrd 2834	. 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 2940  ∀wral 3061   ⊆ wss 3947  ∅c0 4321   ↦ cmpt 5230  ran crn 5676   Fn wfn 6535  ‘cfv 6540  (class class class)co 7405  1c1 11107   + caddc 11109  -cneg 11441  ℤcz 12554  Basecbs 17140  +_gcplusg 17193  Grpcgrp 18815  inv_gcminusg 18816  ._gcmg 18944  SubGrpcsubg 18994

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

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-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-seq 13963  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-mulg 18945  df-subg 18997

This theorem is referenced by:  cycsubg  19079  cycsubgcld  19080  oddvds2  19428  cycsubgcyg  19763