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Theorem lt6abl 18296

Description: A group with fewer than 6 elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)

**Hypothesis**
Ref	Expression
cygctb.1	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
lt6abl	⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐺 ∈ Abel)

Proof of Theorem lt6abl Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cygctb.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
2	1	grpbn0 17451	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
3	2	adantr 481	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐵 ≠ ∅)
4		6re 11101	. . . . . . . 8 ⊢ 6 ∈ ℝ
5		rexr 10085	. . . . . . . 8 ⊢ (6 ∈ ℝ → 6 ∈ ℝ^*)
6		pnfnlt 11962	. . . . . . . 8 ⊢ (6 ∈ ℝ^* → ¬ +∞ < 6)
7	4, 5, 6	mp2b 10	. . . . . . 7 ⊢ ¬ +∞ < 6
8		fvex 6201	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) ∈ V
9	1, 8	eqeltri 2697	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
11		hashinf 13122	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
12	10, 11	sylan 488	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
13	12	breq1d 4663	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐵) < 6 ↔ +∞ < 6))
14	13	biimpd 219	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐵) < 6 → +∞ < 6))
15	14	impancom 456	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (¬ 𝐵 ∈ Fin → +∞ < 6))
16	7, 15	mt3i 141	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐵 ∈ Fin)
17		hashnncl 13157	. . . . . 6 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
19	3, 18	mpbird 247	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) ∈ ℕ)
20		nnuz 11723	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
21	19, 20	syl6eleq 2711	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) ∈ (ℤ_≥‘1))
22		6nn 11189	. . . . 5 ⊢ 6 ∈ ℕ
23	22	nnzi 11401	. . . 4 ⊢ 6 ∈ ℤ
24	23	a1i 11	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 6 ∈ ℤ)
25		simpr 477	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) < 6)
26		elfzo2 12473	. . 3 ⊢ ((#‘𝐵) ∈ (1..^6) ↔ ((#‘𝐵) ∈ (ℤ_≥‘1) ∧ 6 ∈ ℤ ∧ (#‘𝐵) < 6))
27	21, 24, 25, 26	syl3anbrc 1246	. 2 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) ∈ (1..^6))
28		df-6 11083	. . . . . . 7 ⊢ 6 = (5 + 1)
29	28	oveq2i 6661	. . . . . 6 ⊢ (1..^6) = (1..^(5 + 1))
30	29	eleq2i 2693	. . . . 5 ⊢ ((#‘𝐵) ∈ (1..^6) ↔ (#‘𝐵) ∈ (1..^(5 + 1)))
31		5nn 11188	. . . . . . 7 ⊢ 5 ∈ ℕ
32	31, 20	eleqtri 2699	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘1)
33		fzosplitsni 12579	. . . . . 6 ⊢ (5 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(5 + 1)) ↔ ((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5)))
34	32, 33	ax-mp 5	. . . . 5 ⊢ ((#‘𝐵) ∈ (1..^(5 + 1)) ↔ ((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5))
35	30, 34	bitri 264	. . . 4 ⊢ ((#‘𝐵) ∈ (1..^6) ↔ ((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5))
36		df-5 11082	. . . . . . . . 9 ⊢ 5 = (4 + 1)
37	36	oveq2i 6661	. . . . . . . 8 ⊢ (1..^5) = (1..^(4 + 1))
38	37	eleq2i 2693	. . . . . . 7 ⊢ ((#‘𝐵) ∈ (1..^5) ↔ (#‘𝐵) ∈ (1..^(4 + 1)))
39		4nn 11187	. . . . . . . . 9 ⊢ 4 ∈ ℕ
40	39, 20	eleqtri 2699	. . . . . . . 8 ⊢ 4 ∈ (ℤ_≥‘1)
41		fzosplitsni 12579	. . . . . . . 8 ⊢ (4 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(4 + 1)) ↔ ((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4)))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ ((#‘𝐵) ∈ (1..^(4 + 1)) ↔ ((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4))
43	38, 42	bitri 264	. . . . . 6 ⊢ ((#‘𝐵) ∈ (1..^5) ↔ ((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4))
44		df-4 11081	. . . . . . . . . . 11 ⊢ 4 = (3 + 1)
45	44	oveq2i 6661	. . . . . . . . . 10 ⊢ (1..^4) = (1..^(3 + 1))
46	45	eleq2i 2693	. . . . . . . . 9 ⊢ ((#‘𝐵) ∈ (1..^4) ↔ (#‘𝐵) ∈ (1..^(3 + 1)))
47		3nn 11186	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
48	47, 20	eleqtri 2699	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
49		fzosplitsni 12579	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(3 + 1)) ↔ ((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3)))
50	48, 49	ax-mp 5	. . . . . . . . 9 ⊢ ((#‘𝐵) ∈ (1..^(3 + 1)) ↔ ((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3))
51	46, 50	bitri 264	. . . . . . . 8 ⊢ ((#‘𝐵) ∈ (1..^4) ↔ ((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3))
52		df-3 11080	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
53	52	oveq2i 6661	. . . . . . . . . . . 12 ⊢ (1..^3) = (1..^(2 + 1))
54	53	eleq2i 2693	. . . . . . . . . . 11 ⊢ ((#‘𝐵) ∈ (1..^3) ↔ (#‘𝐵) ∈ (1..^(2 + 1)))
55		2eluzge1 11734	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
56		fzosplitsni 12579	. . . . . . . . . . . 12 ⊢ (2 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(2 + 1)) ↔ ((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2)))
57	55, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ ((#‘𝐵) ∈ (1..^(2 + 1)) ↔ ((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2))
58	54, 57	bitri 264	. . . . . . . . . 10 ⊢ ((#‘𝐵) ∈ (1..^3) ↔ ((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2))
59		elsni 4194	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐵) ∈ {1} → (#‘𝐵) = 1)
60		fzo12sn 12551	. . . . . . . . . . . . . . . . 17 ⊢ (1..^2) = {1}
61	59, 60	eleq2s 2719	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐵) ∈ (1..^2) → (#‘𝐵) = 1)
62	61	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → (#‘𝐵) = 1)
63		hash1 13192	. . . . . . . . . . . . . . 15 ⊢ (#‘1_𝑜) = 1
64	62, 63	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → (#‘𝐵) = (#‘1_𝑜))
65		1nn0 11308	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ₀
66	62, 65	syl6eqel 2709	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → (#‘𝐵) ∈ ℕ₀)
67		hashclb 13149	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ V → (𝐵 ∈ Fin ↔ (#‘𝐵) ∈ ℕ₀))
68	9, 67	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin ↔ (#‘𝐵) ∈ ℕ₀)
69	66, 68	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → 𝐵 ∈ Fin)
70		1onn 7719	. . . . . . . . . . . . . . . 16 ⊢ 1_𝑜 ∈ ω
71		nnfi 8153	. . . . . . . . . . . . . . . 16 ⊢ (1_𝑜 ∈ ω → 1_𝑜 ∈ Fin)
72	70, 71	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1_𝑜 ∈ Fin
73		hashen 13135	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ Fin ∧ 1_𝑜 ∈ Fin) → ((#‘𝐵) = (#‘1_𝑜) ↔ 𝐵 ≈ 1_𝑜))
74	69, 72, 73	sylancl 694	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → ((#‘𝐵) = (#‘1_𝑜) ↔ 𝐵 ≈ 1_𝑜))
75	64, 74	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → 𝐵 ≈ 1_𝑜)
76	1	0cyg 18294	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1_𝑜) → 𝐺 ∈ CycGrp)
77		cygabl 18292	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
78	76, 77	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1_𝑜) → 𝐺 ∈ Abel)
79	75, 78	syldan 487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → 𝐺 ∈ Abel)
80	79	ex 450	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^2) → 𝐺 ∈ Abel))
81		id 22	. . . . . . . . . . . . 13 ⊢ ((#‘𝐵) = 2 → (#‘𝐵) = 2)
82		2prm 15405	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℙ
83	81, 82	syl6eqel 2709	. . . . . . . . . . . 12 ⊢ ((#‘𝐵) = 2 → (#‘𝐵) ∈ ℙ)
84	1	prmcyg 18295	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)
85	84, 77	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ ℙ) → 𝐺 ∈ Abel)
86	85	ex 450	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ ℙ → 𝐺 ∈ Abel))
87	83, 86	syl5 34	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 2 → 𝐺 ∈ Abel))
88	80, 87	jaod 395	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2) → 𝐺 ∈ Abel))
89	58, 88	syl5bi 232	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^3) → 𝐺 ∈ Abel))
90		id 22	. . . . . . . . . . 11 ⊢ ((#‘𝐵) = 3 → (#‘𝐵) = 3)
91		3prm 15406	. . . . . . . . . . 11 ⊢ 3 ∈ ℙ
92	90, 91	syl6eqel 2709	. . . . . . . . . 10 ⊢ ((#‘𝐵) = 3 → (#‘𝐵) ∈ ℙ)
93	92, 86	syl5 34	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 3 → 𝐺 ∈ Abel))
94	89, 93	jaod 395	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3) → 𝐺 ∈ Abel))
95	51, 94	syl5bi 232	. . . . . . 7 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^4) → 𝐺 ∈ Abel))
96		simpl 473	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → 𝐺 ∈ Grp)
97		2z 11409	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
98		eqid 2622	. . . . . . . . . . . 12 ⊢ (gEx‘𝐺) = (gEx‘𝐺)
99		eqid 2622	. . . . . . . . . . . 12 ⊢ (od‘𝐺) = (od‘𝐺)
100	1, 98, 99	gexdvds2 18000	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 2 ∈ ℤ) → ((gEx‘𝐺) ∥ 2 ↔ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2))
101	96, 97, 100	sylancl 694	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → ((gEx‘𝐺) ∥ 2 ↔ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2))
102	1, 98	gex2abl 18254	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (gEx‘𝐺) ∥ 2) → 𝐺 ∈ Abel)
103	102	ex 450	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ((gEx‘𝐺) ∥ 2 → 𝐺 ∈ Abel))
104	103	adantr 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → ((gEx‘𝐺) ∥ 2 → 𝐺 ∈ Abel))
105	101, 104	sylbird 250	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2 → 𝐺 ∈ Abel))
106		rexnal 2995	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐵 ¬ ((od‘𝐺)‘𝑥) ∥ 2 ↔ ¬ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2)
107	96	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐺 ∈ Grp)
108		simprl 794	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝑥 ∈ 𝐵)
109	1, 99	odcl 17955	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → ((od‘𝐺)‘𝑥) ∈ ℕ₀)
110	109	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∈ ℕ₀)
111		4nn0 11311	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
112	111	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 4 ∈ ℕ₀)
113		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (#‘𝐵) = 4)
114	113, 111	syl6eqel 2709	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (#‘𝐵) ∈ ℕ₀)
115	114, 68	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → 𝐵 ∈ Fin)
116	115	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐵 ∈ Fin)
117	1, 99	oddvds2 17983	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝑥 ∈ 𝐵) → ((od‘𝐺)‘𝑥) ∥ (#‘𝐵))
118	107, 116, 108, 117	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∥ (#‘𝐵))
119	113	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (#‘𝐵) = 4)
120	118, 119	breqtrd 4679	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∥ 4)
121		sq2 12960	. . . . . . . . . . . . . . . . 17 ⊢ (2↑2) = 4
122	97	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 2 ∈ ℤ)
123		2nn0 11309	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℕ₀
124	123	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 2 ∈ ℕ₀)
125	1, 99	odcl2 17982	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝑥 ∈ 𝐵) → ((od‘𝐺)‘𝑥) ∈ ℕ)
126	107, 116, 108, 125	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∈ ℕ)
127		pccl 15554	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℙ ∧ ((od‘𝐺)‘𝑥) ∈ ℕ) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀)
128	82, 126, 127	sylancr 695	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀)
129	128	nn0zd 11480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℤ)
130		df-2 11079	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 = (1 + 1)
131		simprr 796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ¬ ((od‘𝐺)‘𝑥) ∥ 2)
132		dvdsexp 15049	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2 ∈ ℤ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀ ∧ 1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥)))) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1))
133	132	3expia 1267	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2 ∈ ℤ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1)))
134	97, 128, 133	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1)))
135		1z 11407	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℤ
136		eluz 11701	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℤ ∧ 1 ∈ ℤ) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) ↔ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
137	129, 135, 136	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) ↔ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
138		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 2 → (2↑𝑛) = (2↑2))
139	138, 121	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 2 → (2↑𝑛) = 4)
140	139	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 2 → (((od‘𝐺)‘𝑥) ∥ (2↑𝑛) ↔ ((od‘𝐺)‘𝑥) ∥ 4))
141	140	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℕ₀ ∧ ((od‘𝐺)‘𝑥) ∥ 4) → ∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛))
142	123, 120, 141	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛))
143		pcprmpw2 15586	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℙ ∧ ((od‘𝐺)‘𝑥) ∈ ℕ) → (∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛) ↔ ((od‘𝐺)‘𝑥) = (2↑(2 pCnt ((od‘𝐺)‘𝑥)))))
144	82, 126, 143	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛) ↔ ((od‘𝐺)‘𝑥) = (2↑(2 pCnt ((od‘𝐺)‘𝑥)))))
145	142, 144	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) = (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
146	145	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) = ((od‘𝐺)‘𝑥))
147		2cn 11091	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 2 ∈ ℂ
148		exp1 12866	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2 ∈ ℂ → (2↑1) = 2)
149	147, 148	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2↑1) = 2
150	149	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2↑1) = 2)
151	146, 150	breq12d 4666	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1) ↔ ((od‘𝐺)‘𝑥) ∥ 2))
152	134, 137, 151	3imtr3d 282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1 → ((od‘𝐺)‘𝑥) ∥ 2))
153	131, 152	mtod 189	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ¬ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1)
154		1re 10039	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
155	128	nn0red 11352	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℝ)
156		ltnle 10117	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℝ) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ ¬ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
157	154, 155, 156	sylancr 695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ ¬ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
158	153, 157	mpbird 247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 1 < (2 pCnt ((od‘𝐺)‘𝑥)))
159		nn0ltp1le 11435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℕ₀ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ (1 + 1) ≤ (2 pCnt ((od‘𝐺)‘𝑥))))
160	65, 128, 159	sylancr 695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ (1 + 1) ≤ (2 pCnt ((od‘𝐺)‘𝑥))))
161	158, 160	mpbid 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 + 1) ≤ (2 pCnt ((od‘𝐺)‘𝑥)))
162	130, 161	syl5eqbr 4688	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 2 ≤ (2 pCnt ((od‘𝐺)‘𝑥)))
163		eluz2 11693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 pCnt ((od‘𝐺)‘𝑥)) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℤ ∧ 2 ≤ (2 pCnt ((od‘𝐺)‘𝑥))))
164	122, 129, 162, 163	syl3anbrc 1246	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ (ℤ_≥‘2))
165		dvdsexp 15049	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 2 ∈ ℕ₀ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ (ℤ_≥‘2)) → (2↑2) ∥ (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
166	122, 124, 164, 165	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2↑2) ∥ (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
167	121, 166	syl5eqbrr 4689	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 4 ∥ (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
168	167, 145	breqtrrd 4681	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 4 ∥ ((od‘𝐺)‘𝑥))
169		dvdseq 15036	. . . . . . . . . . . . . . 15 ⊢ (((((od‘𝐺)‘𝑥) ∈ ℕ₀ ∧ 4 ∈ ℕ₀) ∧ (((od‘𝐺)‘𝑥) ∥ 4 ∧ 4 ∥ ((od‘𝐺)‘𝑥))) → ((od‘𝐺)‘𝑥) = 4)
170	110, 112, 120, 168, 169	syl22anc 1327	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) = 4)
171	170, 119	eqtr4d 2659	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) = (#‘𝐵))
172	1, 99, 107, 108, 171	iscygodd 18290	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐺 ∈ CycGrp)
173	172, 77	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐺 ∈ Abel)
174	173	rexlimdvaa 3032	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (∃𝑥 ∈ 𝐵 ¬ ((od‘𝐺)‘𝑥) ∥ 2 → 𝐺 ∈ Abel))
175	106, 174	syl5bir 233	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (¬ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2 → 𝐺 ∈ Abel))
176	105, 175	pm2.61d 170	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → 𝐺 ∈ Abel)
177	176	ex 450	. . . . . . 7 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 4 → 𝐺 ∈ Abel))
178	95, 177	jaod 395	. . . . . 6 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4) → 𝐺 ∈ Abel))
179	43, 178	syl5bi 232	. . . . 5 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^5) → 𝐺 ∈ Abel))
180		id 22	. . . . . . 7 ⊢ ((#‘𝐵) = 5 → (#‘𝐵) = 5)
181		5prm 15815	. . . . . . 7 ⊢ 5 ∈ ℙ
182	180, 181	syl6eqel 2709	. . . . . 6 ⊢ ((#‘𝐵) = 5 → (#‘𝐵) ∈ ℙ)
183	182, 86	syl5 34	. . . . 5 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 5 → 𝐺 ∈ Abel))
184	179, 183	jaod 395	. . . 4 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5) → 𝐺 ∈ Abel))
185	35, 184	syl5bi 232	. . 3 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^6) → 𝐺 ∈ Abel))
186	185	imp 445	. 2 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^6)) → 𝐺 ∈ Abel)
187	27, 186	syldan 487	1 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐺 ∈ Abel)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∅c0 3915 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ωcom 7065 1_𝑜c1o 7553 ≈ cen 7952 Fincfn 7955 ℂcc 9934 ℝcr 9935 1c1 9937 + caddc 9939 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 5c5 11073 6c6 11074 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ..^cfzo 12465 ↑cexp 12860 #chash 13117 ∥ cdvds 14983 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 Grpcgrp 17422 odcod 17944 gExcgex 17945 Abelcabl 18194 CycGrpccyg 18279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-od 17948 df-gex 17949 df-cmn 18195 df-abl 18196 df-cyg 18280

This theorem is referenced by: pgrple2abl 42146

W3C validator