cyc3evpm - Mathbox for Thierry Arnoux

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Theorem cyc3evpm 33693

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 490	. . . . 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 488	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝐷 ∈ Fin)
3		eqid 2761	. . . . . . . . 9 ⊢ (toCyc‘𝐷) = (toCyc‘𝐷)
4		simpr 490	. . . . . . . . . . 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 4150	. . . . . . . . . 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 3641	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} → 𝑢 ∈ Word 𝐷)
7	5, 6	syl 18	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ Word 𝐷)
8		id 23	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝑤 = 𝑢)
9		dmeq 5885	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → dom 𝑤 = dom 𝑢)
10		eqidd 2762	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝐷 = 𝐷)
11	8, 9, 10	f1eq123d 6808	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑢:dom 𝑢–1-1→𝐷))
12	11	elrab 3645	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑢 ∈ Word 𝐷 ∧ 𝑢:dom 𝑢–1-1→𝐷))
13	12	simprbi 503	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} → 𝑢:dom 𝑢–1-1→𝐷)
14	5, 13	syl 18	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢:dom 𝑢–1-1→𝐷)
15		eqid 2761	. . . . . . . . 9 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
16	3, 2, 7, 14, 15	cycpmcl 33659	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)))
17		c0ex 11281	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
18	17	tpid1 4729	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0, 1, 2}
19		fzo0to3tp 13867	. . . . . . . . . . . . . . . . 17 ⊢ (0..^3) = {0, 1, 2}
20	18, 19	eleqtrri 2860	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0..^3)
21	4	elin2d 4151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ (^◡♯ “ {3}))
22		hashf 14462	. . . . . . . . . . . . . . . . . . . 20 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
23		ffn 6701	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
24		elpreima 7049	. . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (^◡♯ “ {3}) → (♯‘𝑢) ∈ {3})
27		elsni 4601	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑢) ∈ {3} → (♯‘𝑢) = 3)
28	21, 26, 27	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (♯‘𝑢) = 3)
29	28	oveq2d 7428	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (0..^(♯‘𝑢)) = (0..^3))
30	20, 29	eleqtrrid 2868	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 0 ∈ (0..^(♯‘𝑢)))
31		wrdsymbcl 14652	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 0 ∈ (0..^(♯‘𝑢))) → (𝑢‘0) ∈ 𝐷)
32	7, 30, 31	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ∈ 𝐷)
33		1eltp012 12394	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ {0, 1, 2}
34	33, 19	eleqtrri 2860	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0..^3)
35	34, 29	eleqtrrid 2868	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 1 ∈ (0..^(♯‘𝑢)))
36		wrdsymbcl 14652	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 1 ∈ (0..^(♯‘𝑢))) → (𝑢‘1) ∈ 𝐷)
37	7, 35, 36	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) ∈ 𝐷)
38		2ex 12401	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
39	38	tpid3 4734	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ {0, 1, 2}
40	39, 19	eleqtrri 2860	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ (0..^3)
41	40, 29	eleqtrrid 2868	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 2 ∈ (0..^(♯‘𝑢)))
42		wrdsymbcl 14652	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝐷 ∧ 2 ∈ (0..^(♯‘𝑢))) → (𝑢‘2) ∈ 𝐷)
43	7, 41, 42	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ∈ 𝐷)
44	32, 37, 43	3jca 1146	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷))
45		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) = (𝑢‘0))
46		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) = (𝑢‘1))
47		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) = (𝑢‘2))
48	45, 46, 47	3jca 1146	. . . . . . . . . . . . 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)))
49		eqwrds3 15094	. . . . . . . . . . . . . 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)))))
50	49	biimpar 483	. . . . . . . . . . . . 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)”⟩)
51	7, 44, 28, 48, 50	syl22anc 852	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 = ⟨“(𝑢‘0)(𝑢‘1)(𝑢‘2)”⟩)
52	51	fveq2d 6881	. . . . . . . . . . 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)”⟩))
53		wrddm 14646	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ Word 𝐷 → dom 𝑢 = (0..^(♯‘𝑢)))
54	7, 53	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^(♯‘𝑢)))
55	54, 29	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^3))
56	55, 19	eqtrdi 2812	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = {0, 1, 2})
57		f1eq2 6766	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑢 = {0, 1, 2} → (𝑢:dom 𝑢–1-1→𝐷 ↔ 𝑢:{0, 1, 2}–1-1→𝐷))
58	57	biimpa 482	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑢 = {0, 1, 2} ∧ 𝑢:dom 𝑢–1-1→𝐷) → 𝑢:{0, 1, 2}–1-1→𝐷)
59	56, 14, 58	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢:{0, 1, 2}–1-1→𝐷)
60		1ex 11284	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
61	17, 60, 38	3pm3.2i 1358	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V)
62		0ne1 12395	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
63		0ne2 12533	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 2
64		1ne2 12534	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 2
65	62, 63, 64	3pm3.2i 1358	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)
66		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ {0, 1, 2} = {0, 1, 2}
67	66	f13dfv 7274	. . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . 14 ⊢ (𝑢:{0, 1, 2}–1-1→𝐷 → ((𝑢‘0) ≠ (𝑢‘1) ∧ (𝑢‘0) ≠ (𝑢‘2) ∧ (𝑢‘1) ≠ (𝑢‘2)))
70	59, 69	syl 18	. . . . . . . . . . . . 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 1160	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ≠ (𝑢‘1))
72	70	simp3d 1162	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) ≠ (𝑢‘2))
73	70	simp2d 1161	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) ≠ (𝑢‘2))
74	73	necomd 3011	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ≠ (𝑢‘0))
75		eqid 2761	. . . . . . . . . . . 12 ⊢ (+_g‘(SymGrp‘𝐷)) = (+_g‘(SymGrp‘𝐷))
76	3, 15, 2, 32, 37, 43, 71, 72, 74, 75	cyc3co2 33683	. . . . . . . . . . 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, 43, 73, 15	cycpm2cl 33663	. . . . . . . . . . . 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, 37, 71, 15	cycpm2cl 33663	. . . . . . . . . . . 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 2761	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
80	15, 79, 75	symgov 19578	. . . . . . . . . . . 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 596	. . . . . . . . . . 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	52, 76, 81	3eqtrd 2800	. . . . . . . . . 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 6881	. . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
85	15, 84, 79	psgnco 21869	. . . . . . . . . 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 1398	. . . . . . . . 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 2761	. . . . . . . . . . . . . 14 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 2, 32, 43, 73, 87	cycpm2tr 33662	. . . . . . . . . . . . 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, 43	prssd 4783	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ⊆ 𝐷)
90		enpr2 10064	. . . . . . . . . . . . . . 15 ⊢ (((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘2) ∈ 𝐷 ∧ (𝑢‘0) ≠ (𝑢‘2)) → {(𝑢‘0), (𝑢‘2)} ≈ 2_o)
91	32, 43, 73, 90	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ≈ 2_o)
92		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
93	87, 92	pmtrrn 19651	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 19704	. . . . . . . . . . . 12 ⊢ (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩) ∈ ran (pmTrsp‘𝐷) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘2)”⟩)) = -1)
97	95, 96	syl 18	. . . . . . . . . . 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, 37, 71, 87	cycpm2tr 33662	. . . . . . . . . . . . 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, 37	prssd 4783	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ⊆ 𝐷)
100		enpr2 10064	. . . . . . . . . . . . . . 15 ⊢ (((𝑢‘0) ∈ 𝐷 ∧ (𝑢‘1) ∈ 𝐷 ∧ (𝑢‘0) ≠ (𝑢‘1)) → {(𝑢‘0), (𝑢‘1)} ≈ 2_o)
101	32, 37, 71, 100	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ≈ 2_o)
102	87, 92	pmtrrn 19651	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 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 19704	. . . . . . . . . . . 12 ⊢ (((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩) ∈ ran (pmTrsp‘𝐷) → ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘⟨“(𝑢‘0)(𝑢‘1)”⟩)) = -1)
106	104, 105	syl 18	. . . . . . . . . . 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 7430	. . . . . . . . . 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 12539	. . . . . . . . . 10 ⊢ (-1 · -1) = 1
109	107, 108	eqtrdi 2812	. . . . . . . . 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 2800	. . . . . . . 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 21873	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (((toCyc‘𝐷)‘𝑢) ∈ (pmEven‘𝐷) ↔ (((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)) ∧ ((pmSgn‘𝐷)‘((toCyc‘𝐷)‘𝑢)) = 1)))
112	111	biimpar 483	. . . . . . . 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 850	. . . . . . 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 2872	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ 𝐴)
116	115	ad4ant13 764	. . . . 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 2862	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → 𝑝 ∈ 𝐴)
118		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑢(toCyc‘𝐷)
119	3, 15, 79	tocycf 33660	. . . . . . 7 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷):{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘(SymGrp‘𝐷)))
120	119	ffnd 6702	. . . . . 6 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷) Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
121	120	adantr 486	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → (toCyc‘𝐷) Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
122		simpr 490	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐶)
123		cyc3evpm.t	. . . . . 6 ⊢ 𝐶 = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
124	122, 123	eleqtrdi 2871	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ ((toCyc‘𝐷) “ (^◡♯ “ {3})))
125	118, 121, 124	fvelimad 6944	. . . 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 3171	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐴)
127	126	ex 418	. 2 ⊢ (𝐷 ∈ Fin → (𝑝 ∈ 𝐶 → 𝑝 ∈ 𝐴))
128	127	ssrdv 3937	1 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 {crab 3413 Vcvv 3451 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 {csn 4584 {cpr 4586 {ctp 4588 class class class wbr 5103 ^◡ccnv 5650 dom cdm 5651 ran crn 5652 “ cima 5654 ∘ ccom 5655 Fn wfn 6526 ⟶wf 6527 –1-1→wf1 6528 ‘cfv 6531 (class class class)co 7412 2_oc2o 8454 ≈ cen 8954 Fincfn 8957 0cc0 11181 1c1 11182 · cmul 11186 +∞cpnf 11321 -cneg 11523 2c2 12378 3c3 12379 ℕ₀cn0 12587 ..^cfzo 13768 ♯chash 14454 Word cword 14638 ⟨“cs2 14972 ⟨“cs3 14973 Basecbs 17367 +_gcplusg 17408 SymGrpcsymg 19563 pmTrspcpmtr 19635 pmSgncpsgn 19683 pmEvencevpm 19684 toCycctocyc 33649

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260 ax-mulf 11261

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9418 df-inf 9419 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-xnn0 12661 df-z 12675 df-dec 12796 df-uz 12947 df-rp 13102 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-hash 14455 df-word 14639 df-lsw 14688 df-concat 14696 df-s1 14723 df-substr 14769 df-pfx 14801 df-splice 14879 df-reverse 14888 df-csh 14920 df-s2 14979 df-s3 14980 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-0g 17592 df-gsum 17593 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-efmnd 19045 df-grp 19127 df-minusg 19128 df-subg 19313 df-ghm 19408 df-gim 19453 df-oppg 19540 df-symg 19564 df-pmtr 19636 df-psgn 19685 df-evpm 19686 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-cring 20442 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-drng 20962 df-cnfld 21659 df-tocyc 33650

This theorem is used by: cyc3genpm 33695

W3C validator