Theorem cycsubgcl 19189

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 19074	. . . . . . 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 652	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝐴) ∈ 𝑋)
6		cycsubg.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))
7	5, 6	fmptd 7104	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹:ℤ⟶𝑋)
8	7	frnd 6714	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ⊆ 𝑋)
9	7	ffnd 6707	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn ℤ)
10		1z 12622	. . . . 5 ⊢ 1 ∈ ℤ
11		fnfvelrn 7070	. . . . 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 4317	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ≠ ∅)
14		df-3an 1088	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐴 ∈ 𝑋) ↔ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝐴 ∈ 𝑋))
15		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
16	1, 2, 15	mulgdir 19089	. . . . . . . . . . . . 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 655	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴)(+g‘𝐺)(𝑛 · 𝐴)))
19		zaddcl 12632	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + 𝑛) ∈ ℤ)
20	19	adantl 481	. . . . . . . . . . . 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 6986	. . . . . . . . . . . 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 6986	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℤ → (𝐹‘𝑚) = (𝑚 · 𝐴))
28	27	ad2antrl 728	. . . . . . . . . . . 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 6986	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝐹‘𝑛) = (𝑛 · 𝐴))
32	31	ad2antll 729	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → (𝐹‘(𝑚 + 𝑛)) = ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)))
35		fnfvelrn 7070	. . . . . . . . . . 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 2835	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹)
38	37	anassrs 467	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) ∧ 𝑛 ∈ ℤ) → ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹)
39	38	ralrimiva 3132	. . . . . . 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 2819	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑛) → (((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹‘𝑚)(+g‘𝐺)(𝐹‘𝑛)) ∈ ran 𝐹))
42	41	ralrn 7078	. . . . . . . . 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 480	. . . . . . 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 257	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ∀𝑣 ∈ ran 𝐹((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹)
46		eqid 2735	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
47	1, 2, 46	mulgneg 19075	. . . . . . . . . 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 652	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (-𝑚 · 𝐴) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
50		znegcl 12627	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → -𝑚 ∈ ℤ)
51	50	adantl 481	. . . . . . . . 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 6986	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) = (𝑚 · 𝐴))
57	56	fveq2d 6880	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ((invg‘𝐺)‘(𝐹‘𝑚)) = ((invg‘𝐺)‘(𝑚 · 𝐴)))
58	49, 55, 57	3eqtr4d 2780	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → (𝐹‘-𝑚) = ((invg‘𝐺)‘(𝐹‘𝑚)))
59		fnfvelrn 7070	. . . . . . . 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 2835	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) ∧ 𝑚 ∈ ℤ) → ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹)
62	45, 61	jca 511	. . . . 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 3132	. . . 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 2819	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑚) → ((𝑢(+g‘𝐺)𝑣) ∈ ran 𝐹 ↔ ((𝐹‘𝑚)(+g‘𝐺)𝑣) ∈ ran 𝐹))
66	65	ralbidv 3163	. . . . . . 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 6876	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑚) → ((invg‘𝐺)‘𝑢) = ((invg‘𝐺)‘(𝐹‘𝑚)))
68	67	eleq1d 2819	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑚) → (((invg‘𝐺)‘𝑢) ∈ ran 𝐹 ↔ ((invg‘𝐺)‘(𝐹‘𝑚)) ∈ ran 𝐹))
69	66, 68	anbi12d 632	. . . . . 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 7078	. . . . 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 257	. . 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 19124	. . . 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 480	. . 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 1343	. 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 6986	. . . . 5 ⊢ (1 ∈ ℤ → (𝐹‘1) = (1 · 𝐴))
79	10, 78	ax-mp 5	. . . 4 ⊢ (𝐹‘1) = (1 · 𝐴)
80	1, 2	mulg1 19064	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (1 · 𝐴) = 𝐴)
81	80	adantl 481	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (1 · 𝐴) = 𝐴)
82	79, 81	eqtrid 2782	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹‘1) = 𝐴)
83	82, 12	eqeltrrd 2835	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ran 𝐹)
84	75, 83	jca 511	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051   ⊆ wss 3926  ∅c0 4308   ↦ cmpt 5201  ran crn 5655   Fn wfn 6526  ‘cfv 6531  (class class class)co 7405  1c1 11130   + caddc 11132  -cneg 11467  ℤcz 12588  Basecbs 17228  +_gcplusg 17271  Grpcgrp 18916  inv_gcminusg 18917  ._gcmg 19050  SubGrpcsubg 19103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-seq 14020  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-mulg 19051  df-subg 19106

This theorem is referenced by:  cycsubg  19191  cycsubgcld  19192  oddvds2  19547  cycsubgcyg  19882