cyc3evpm - Mathbox for Thierry Arnoux

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Theorem cyc3evpm 33143

Description: 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)

**Hypotheses**
Ref	Expression
cyc3evpm.t	⊢ 𝐶 = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
cyc3evpm.a	⊢ 𝐴 = (pmEven‘𝐷)

**Assertion**
Ref	Expression
cyc3evpm	⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)

Proof of Theorem cyc3evpm Dummy variables 𝑝 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → ((toCyc‘𝐷)‘𝑢) = 𝑝)
2		simpl 482	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝐷 ∈ Fin)
3		eqid 2740	. . . . . . . . 9 ⊢ (toCyc‘𝐷) = (toCyc‘𝐷)
4		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3})))
5	4	elin1d 4227	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
6		elrabi 3703	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} → 𝑢 ∈ Word 𝐷)
7	5, 6	syl 17	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ Word 𝐷)
8		id 22	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝑤 = 𝑢)
9		dmeq 5928	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → dom 𝑤 = dom 𝑢)
10		eqidd 2741	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝐷 = 𝐷)
11	8, 9, 10	f1eq123d 6854	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑢:dom 𝑢–1-1→𝐷))
12	11	elrab 3708	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑢 ∈ Word 𝐷 ∧ 𝑢:dom 𝑢–1-1→𝐷))
13	12	simprbi 496	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} → 𝑢:dom 𝑢–1-1→𝐷)
14	5, 13	syl 17	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢:dom 𝑢–1-1→𝐷)
15		eqid 2740	. . . . . . . . 9 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
16	3, 2, 7, 14, 15	cycpmcl 33109	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)))
17		c0ex 11284	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
18	17	tpid1 4793	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0, 1, 2}
19		fzo0to3tp 13802	. . . . . . . . . . . . . . . . 17 ⊢ (0..^3) = {0, 1, 2}
20	18, 19	eleqtrri 2843	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0..^3)
21	4	elin2d 4228	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ (^◡♯ “ {3}))
22		hashf 14387	. . . . . . . . . . . . . . . . . . . 20 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
23		ffn 6747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
24		elpreima 7091	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯ Fn V → (𝑢 ∈ (^◡♯ “ {3}) ↔ (𝑢 ∈ V ∧ (♯‘𝑢) ∈ {3})))
25	22, 23, 24	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (^◡♯ “ {3}) ↔ (𝑢 ∈ V ∧ (♯‘𝑢) ∈ {3}))
26	25	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (^◡♯ “ {3}) → (♯‘𝑢) ∈ {3})
27		elsni 4665	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑢) ∈ {3} → (♯‘𝑢) = 3)
28	21, 26, 27	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (♯‘𝑢) = 3)
29	28	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (0..^(♯‘𝑢)) = (0..^3))
30	20, 29	eleqtrrid 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 0 ∈ (0..^(♯‘𝑢)))
31		wrdsymbcl 14575	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 0 ∈ (0..^(♯‘𝑢))) → (𝑢‘0) ∈ 𝐷)
32	7, 30, 31	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ∈ 𝐷)
33		1ex 11286	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
34	33	tpid2 4795	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ {0, 1, 2}
35	34, 19	eleqtrri 2843	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0..^3)
36	35, 29	eleqtrrid 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 1 ∈ (0..^(♯‘𝑢)))
37		wrdsymbcl 14575	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 1 ∈ (0..^(♯‘𝑢))) → (𝑢‘1) ∈ 𝐷)
38	7, 36, 37	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) ∈ 𝐷)
39		2ex 12370	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40	39	tpid3 4798	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ {0, 1, 2}
41	40, 19	eleqtrri 2843	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ (0..^3)
42	41, 29	eleqtrrid 2851	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 2 ∈ (0..^(♯‘𝑢)))
43		wrdsymbcl 14575	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 2 ∈ (0..^(♯‘𝑢))) → (𝑢‘2) ∈ 𝐷)
44	7, 42, 43	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ∈ 𝐷)
45	32, 38, 44	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷))
46		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) = (𝑢‘0))
47		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) = (𝑢‘1))
48		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) = (𝑢‘2))
49	46, 47, 48	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((𝑢‘0) = (𝑢‘0) ∧ (𝑢‘1) = (𝑢‘1) ∧ (𝑢‘2) = (𝑢‘2)))
50		eqwrds3 15010	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Word 𝐷 ∧ ((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷)) → (𝑢 = ⟨“(𝑢‘0)(𝑢‘1)(𝑢‘2)”⟩ ↔ ((♯‘𝑢) = 3 ∧ ((𝑢‘0) = (𝑢‘0) ∧ (𝑢‘1) = (𝑢‘1) ∧ (𝑢‘2) = (𝑢‘2)))))
51	50	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ Word 𝐷 ∧ ((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷)) ∧ ((♯‘𝑢) = 3 ∧ ((𝑢‘0) = (𝑢‘0) ∧ (𝑢‘1) = (𝑢‘1) ∧ (𝑢‘2) = (𝑢‘2)))) → 𝑢 = ⟨“(𝑢‘0)(𝑢‘1)(𝑢‘2)”⟩)
52	7, 45, 28, 49, 51	syl22anc 838	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 = ⟨“(𝑢‘0)(𝑢‘1)(𝑢‘2)”⟩)
53	52	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) = ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)(𝑢‘2)”⟩))
54		wrddm 14569	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ Word 𝐷 → dom 𝑢 = (0..^(♯‘𝑢)))
55	7, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^(♯‘𝑢)))
56	55, 29	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^3))
57	56, 19	eqtrdi 2796	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = {0, 1, 2})
58		f1eq2 6813	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑢 = {0, 1, 2} → (𝑢:dom 𝑢–1-1→𝐷 ↔ 𝑢:{0, 1, 2}–1-1→𝐷))
59	58	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑢 = {0, 1, 2} ∧ 𝑢:dom 𝑢–1-1→𝐷) → 𝑢:{0, 1, 2}–1-1→𝐷)
60	57, 14, 59	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢:{0, 1, 2}–1-1→𝐷)
61	17, 33, 39	3pm3.2i 1339	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V)
62		0ne1 12364	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
63		0ne2 12500	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 2
64		1ne2 12501	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 2
65	62, 63, 64	3pm3.2i 1339	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)
66		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ {0, 1, 2} = {0, 1, 2}
67	66	f13dfv 7310	. . . . . . . . . . . . . . . 16 ⊢ (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V) ∧ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)) → (𝑢:{0, 1, 2}–1-1→𝐷 ↔ (𝑢:{0, 1, 2}⟶𝐷 ∧ ((𝑢‘0) ≠ (𝑢‘1) ∧ (𝑢‘0) ≠ (𝑢‘2) ∧ (𝑢‘1) ≠ (𝑢‘2)))))
68	61, 65, 67	mp2an 691	. . . . . . . . . . . . . . 15 ⊢ (𝑢:{0, 1, 2}–1-1→𝐷 ↔ (𝑢:{0, 1, 2}⟶𝐷 ∧ ((𝑢‘0) ≠ (𝑢‘1) ∧ (𝑢‘0) ≠ (𝑢‘2) ∧ (𝑢‘1) ≠ (𝑢‘2))))
69	68	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑢:{0, 1, 2}–1-1→𝐷 → ((𝑢‘0) ≠ (𝑢‘1) ∧ (𝑢‘0) ≠ (𝑢‘2) ∧ (𝑢‘1) ≠ (𝑢‘2)))
70	60, 69	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((𝑢‘0) ≠ (𝑢‘1) ∧ (𝑢‘0) ≠ (𝑢‘2) ∧ (𝑢‘1) ≠ (𝑢‘2)))
71	70	simp1d 1142	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ≠ (𝑢‘1))
72	70	simp3d 1144	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) ≠ (𝑢‘2))
73	70	simp2d 1143	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ≠ (𝑢‘2))
74	73	necomd 3002	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ≠ (𝑢‘0))
75		eqid 2740	. . . . . . . . . . . 12 ⊢ (+_g‘(SymGrp‘𝐷)) = (+_g‘(SymGrp‘𝐷))
76	3, 15, 2, 32, 38, 44, 71, 72, 74, 75	cyc3co2 33133	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)(𝑢‘2)”⟩) = (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)(+_g‘(SymGrp‘𝐷))((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)))
77	3, 2, 32, 44, 73, 15	cycpm2cl 33113	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∈ (Base‘(SymGrp‘𝐷)))
78	3, 2, 32, 38, 71, 15	cycpm2cl 33113	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) ∈ (Base‘(SymGrp‘𝐷)))
79		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
80	15, 79, 75	symgov 19425	. . . . . . . . . . . 12 ⊢ ((((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∈ (Base‘(SymGrp‘𝐷)) ∧ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) ∈ (Base‘(SymGrp‘𝐷))) → (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)(+_g‘(SymGrp‘𝐷))((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)) = (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∘ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)))
81	77, 78, 80	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)(+_g‘(SymGrp‘𝐷))((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)) = (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∘ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)))
82	53, 76, 81	3eqtrd 2784	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) = (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∘ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)))
83	82	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘𝑢)) = ((pmSgn‘𝐷)‘(((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∘ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))))
84		eqid 2740	. . . . . . . . . . 11 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
85	15, 84, 79	psgnco 21624	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∈ (Base‘(SymGrp‘𝐷)) ∧ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) ∈ (Base‘(SymGrp‘𝐷))) → ((pmSgn‘𝐷)‘(((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∘ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))) = (((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) · ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))))
86	2, 77, 78, 85	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmSgn‘𝐷)‘(((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∘ ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))) = (((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) · ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))))
87		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 2, 32, 44, 73, 87	cycpm2tr 33112	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) = ((pmTrsp‘𝐷)‘{(𝑢‘0), (𝑢‘2)}))
89	32, 44	prssd 4847	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ⊆ 𝐷)
90		enpr2 10071	. . . . . . . . . . . . . . 15 ⊢ (((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷 ∧ (𝑢‘0) ≠ (𝑢‘2)) → {(𝑢‘0), (𝑢‘2)} ≈ 2_o)
91	32, 44, 73, 90	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ≈ 2_o)
92		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
93	87, 92	pmtrrn 19499	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ {(𝑢‘0), (𝑢‘2)} ⊆ 𝐷 ∧ {(𝑢‘0), (𝑢‘2)} ≈ 2_o) → ((pmTrsp‘𝐷)‘{(𝑢‘0), (𝑢‘2)}) ∈ ran (pmTrsp‘𝐷))
94	2, 89, 91, 93	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmTrsp‘𝐷)‘{(𝑢‘0), (𝑢‘2)}) ∈ ran (pmTrsp‘𝐷))
95	88, 94	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∈ ran (pmTrsp‘𝐷))
96	15, 92, 84	psgnpmtr 19552	. . . . . . . . . . . 12 ⊢ (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∈ ran (pmTrsp‘𝐷) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) = -1)
97	95, 96	syl 17	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) = -1)
98	3, 2, 32, 38, 71, 87	cycpm2tr 33112	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) = ((pmTrsp‘𝐷)‘{(𝑢‘0), (𝑢‘1)}))
99	32, 38	prssd 4847	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ⊆ 𝐷)
100		enpr2 10071	. . . . . . . . . . . . . . 15 ⊢ (((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘0) ≠ (𝑢‘1)) → {(𝑢‘0), (𝑢‘1)} ≈ 2_o)
101	32, 38, 71, 100	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ≈ 2_o)
102	87, 92	pmtrrn 19499	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ {(𝑢‘0), (𝑢‘1)} ⊆ 𝐷 ∧ {(𝑢‘0), (𝑢‘1)} ≈ 2_o) → ((pmTrsp‘𝐷)‘{(𝑢‘0), (𝑢‘1)}) ∈ ran (pmTrsp‘𝐷))
103	2, 99, 101, 102	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmTrsp‘𝐷)‘{(𝑢‘0), (𝑢‘1)}) ∈ ran (pmTrsp‘𝐷))
104	98, 103	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) ∈ ran (pmTrsp‘𝐷))
105	15, 92, 84	psgnpmtr 19552	. . . . . . . . . . . 12 ⊢ (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) ∈ ran (pmTrsp‘𝐷) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)) = -1)
106	104, 105	syl 17	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)) = -1)
107	97, 106	oveq12d 7466	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) · ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))) = (-1 · -1))
108		neg1mulneg1e1 12506	. . . . . . . . . 10 ⊢ (-1 · -1) = 1
109	107, 108	eqtrdi 2796	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) · ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩))) = 1)
110	83, 86, 109	3eqtrd 2784	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘𝑢)) = 1)
111	15, 79, 84	psgnevpmb 21628	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (((toCyc‘𝐷)‘𝑢) ∈ (pmEven‘𝐷) ↔ (((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)) ∧ ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘𝑢)) = 1)))
112	111	biimpar 477	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ (((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)) ∧ ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘𝑢)) = 1)) → ((toCyc‘𝐷)‘𝑢) ∈ (pmEven‘𝐷))
113	2, 16, 110, 112	syl12anc 836	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ (pmEven‘𝐷))
114		cyc3evpm.a	. . . . . . 7 ⊢ 𝐴 = (pmEven‘𝐷)
115	113, 114	eleqtrrdi 2855	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ 𝐴)
116	115	ad4ant13 750	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → ((toCyc‘𝐷)‘𝑢) ∈ 𝐴)
117	1, 116	eqeltrrd 2845	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → 𝑝 ∈ 𝐴)
118		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑢(toCyc‘𝐷)
119	3, 15, 79	tocycf 33110	. . . . . . 7 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷):{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘(SymGrp‘𝐷)))
120	119	ffnd 6748	. . . . . 6 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷) Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
121	120	adantr 480	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → (toCyc‘𝐷) Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
122		simpr 484	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐶)
123		cyc3evpm.t	. . . . . 6 ⊢ 𝐶 = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
124	122, 123	eleqtrdi 2854	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ ((toCyc‘𝐷) “ (^◡♯ “ {3})))
125	118, 121, 124	fvelimad 6989	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → ∃𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))((toCyc‘𝐷)‘𝑢) = 𝑝)
126	117, 125	r19.29a 3168	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐴)
127	126	ex 412	. 2 ⊢ (𝐷 ∈ Fin → (𝑝 ∈ 𝐶 → 𝑝 ∈ 𝐴))
128	127	ssrdv 4014	1 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 {crab 3443 Vcvv 3488 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 {csn 4648 {cpr 4650 {ctp 4652 class class class wbr 5166 ^◡ccnv 5699 dom cdm 5700 ran crn 5701 “ cima 5703 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 –1-1→wf1 6570 ‘cfv 6573 (class class class)co 7448 2_oc2o 8516 ≈ cen 9000 Fincfn 9003 0cc0 11184 1c1 11185 · cmul 11189 +∞cpnf 11321 -cneg 11521 2c2 12348 3c3 12349 ℕ₀cn0 12553 ..^cfzo 13711 ♯chash 14379 Word cword 14562 ⟨“cs2 14890 ⟨“cs3 14891 Basecbs 17258 +_gcplusg 17311 SymGrpcsymg 19410 pmTrspcpmtr 19483 pmSgncpsgn 19531 pmEvencevpm 19532 toCycctocyc 33099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-word 14563 df-lsw 14611 df-concat 14619 df-s1 14644 df-substr 14689 df-pfx 14719 df-splice 14798 df-reverse 14807 df-csh 14837 df-s2 14897 df-s3 14898 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-efmnd 18904 df-grp 18976 df-minusg 18977 df-subg 19163 df-ghm 19253 df-gim 19299 df-oppg 19386 df-symg 19411 df-pmtr 19484 df-psgn 19533 df-evpm 19534 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-cnfld 21388 df-tocyc 33100

This theorem is referenced by: cyc3genpm 33145

W3C validator