cyc3evpm - Mathbox for Thierry Arnoux

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Theorem cyc3evpm 33105

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 2729	. . . . . . . . 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 4157	. . . . . . . . . 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 3645	. . . . . . . . . 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 5850	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → dom 𝑤 = dom 𝑢)
10		eqidd 2730	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝐷 = 𝐷)
11	8, 9, 10	f1eq123d 6760	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑢:dom 𝑢–1-1→𝐷))
12	11	elrab 3650	. . . . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
16	3, 2, 7, 14, 15	cycpmcl 33071	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)))
17		c0ex 11128	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
18	17	tpid1 4722	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0, 1, 2}
19		fzo0to3tp 13673	. . . . . . . . . . . . . . . . 17 ⊢ (0..^3) = {0, 1, 2}
20	18, 19	eleqtrri 2827	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0..^3)
21	4	elin2d 4158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ (^◡♯ “ {3}))
22		hashf 14263	. . . . . . . . . . . . . . . . . . . 20 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
23		ffn 6656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
24		elpreima 6996	. . . . . . . . . . . . . . . . . . . 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 4596	. . . . . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (0..^(♯‘𝑢)) = (0..^3))
30	20, 29	eleqtrrid 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 0 ∈ (0..^(♯‘𝑢)))
31		wrdsymbcl 14452	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 0 ∈ (0..^(♯‘𝑢))) → (𝑢‘0) ∈ 𝐷)
32	7, 30, 31	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ∈ 𝐷)
33		1ex 11130	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
34	33	tpid2 4724	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ {0, 1, 2}
35	34, 19	eleqtrri 2827	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0..^3)
36	35, 29	eleqtrrid 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 1 ∈ (0..^(♯‘𝑢)))
37		wrdsymbcl 14452	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 1 ∈ (0..^(♯‘𝑢))) → (𝑢‘1) ∈ 𝐷)
38	7, 36, 37	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) ∈ 𝐷)
39		2ex 12223	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40	39	tpid3 4727	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ {0, 1, 2}
41	40, 19	eleqtrri 2827	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ (0..^3)
42	41, 29	eleqtrrid 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 2 ∈ (0..^(♯‘𝑢)))
43		wrdsymbcl 14452	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 2 ∈ (0..^(♯‘𝑢))) → (𝑢‘2) ∈ 𝐷)
44	7, 42, 43	syl2anc 584	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) = (𝑢‘0))
47		eqidd 2730	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) = (𝑢‘1))
48		eqidd 2730	. . . . . . . . . . . . . 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 14886	. . . . . . . . . . . . . 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 6830	. . . . . . . . . . 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 14446	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^3))
57	56, 19	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = {0, 1, 2})
58		f1eq2 6720	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 1340	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V)
62		0ne1 12217	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
63		0ne2 12348	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 2
64		1ne2 12349	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 2
65	62, 63, 64	3pm3.2i 1340	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)
66		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ {0, 1, 2} = {0, 1, 2}
67	66	f13dfv 7215	. . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . 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 2980	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ≠ (𝑢‘0))
75		eqid 2729	. . . . . . . . . . . 12 ⊢ (+_g‘(SymGrp‘𝐷)) = (+_g‘(SymGrp‘𝐷))
76	3, 15, 2, 32, 38, 44, 71, 72, 74, 75	cyc3co2 33095	. . . . . . . . . . 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 33075	. . . . . . . . . . . 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 33075	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
80	15, 79, 75	symgov 19281	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2768	. . . . . . . . . 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 6830	. . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
85	15, 84, 79	psgnco 21508	. . . . . . . . . 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 1373	. . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 2, 32, 44, 73, 87	cycpm2tr 33074	. . . . . . . . . . . . 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 4776	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ⊆ 𝐷)
90		enpr2 9917	. . . . . . . . . . . . . . 15 ⊢ (((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷 ∧ (𝑢‘0) ≠ (𝑢‘2)) → {(𝑢‘0), (𝑢‘2)} ≈ 2_o)
91	32, 44, 73, 90	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ≈ 2_o)
92		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
93	87, 92	pmtrrn 19354	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . 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 19407	. . . . . . . . . . . 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 33074	. . . . . . . . . . . . 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 4776	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ⊆ 𝐷)
100		enpr2 9917	. . . . . . . . . . . . . . 15 ⊢ (((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘0) ≠ (𝑢‘1)) → {(𝑢‘0), (𝑢‘1)} ≈ 2_o)
101	32, 38, 71, 100	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ≈ 2_o)
102	87, 92	pmtrrn 19354	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . 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 19407	. . . . . . . . . . . 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 7371	. . . . . . . . . 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 12354	. . . . . . . . . 10 ⊢ (-1 · -1) = 1
109	107, 108	eqtrdi 2780	. . . . . . . . 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 2768	. . . . . . . 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 21512	. . . . . . . . 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 2839	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ 𝐴)
116	115	ad4ant13 751	. . . . 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 2829	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → 𝑝 ∈ 𝐴)
118		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑢(toCyc‘𝐷)
119	3, 15, 79	tocycf 33072	. . . . . . 7 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷):{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘(SymGrp‘𝐷)))
120	119	ffnd 6657	. . . . . 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 2838	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ ((toCyc‘𝐷) “ (^◡♯ “ {3})))
125	118, 121, 124	fvelimad 6894	. . . 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 3137	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐴)
127	126	ex 412	. 2 ⊢ (𝐷 ∈ Fin → (𝑝 ∈ 𝐶 → 𝑝 ∈ 𝐴))
128	127	ssrdv 3943	1 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3396 Vcvv 3438 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 {csn 4579 {cpr 4581 {ctp 4583 class class class wbr 5095 ^◡ccnv 5622 dom cdm 5623 ran crn 5624 “ cima 5626 ∘ ccom 5627 Fn wfn 6481 ⟶wf 6482 –1-1→wf1 6483 ‘cfv 6486 (class class class)co 7353 2_oc2o 8389 ≈ cen 8876 Fincfn 8879 0cc0 11028 1c1 11029 · cmul 11033 +∞cpnf 11165 -cneg 11366 2c2 12201 3c3 12202 ℕ₀cn0 12402 ..^cfzo 13575 ♯chash 14255 Word cword 14438 ⟨“cs2 14766 ⟨“cs3 14767 Basecbs 17138 +_gcplusg 17179 SymGrpcsymg 19266 pmTrspcpmtr 19338 pmSgncpsgn 19386 pmEvencevpm 19387 toCycctocyc 33061

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12610 df-uz 12754 df-rp 12912 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-hash 14256 df-word 14439 df-lsw 14488 df-concat 14496 df-s1 14521 df-substr 14566 df-pfx 14596 df-splice 14674 df-reverse 14683 df-csh 14713 df-s2 14773 df-s3 14774 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17363 df-gsum 17364 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-efmnd 18761 df-grp 18833 df-minusg 18834 df-subg 19020 df-ghm 19110 df-gim 19156 df-oppg 19243 df-symg 19267 df-pmtr 19339 df-psgn 19388 df-evpm 19389 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-drng 20634 df-cnfld 21280 df-tocyc 33062

This theorem is referenced by: cyc3genpm 33107

W3C validator