cyc3evpm - Mathbox for Thierry Arnoux

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Theorem cyc3evpm 33501

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 2766	. . . . . . . . 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 4160	. . . . . . . . . 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 3649	. . . . . . . . . 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 5898	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → dom 𝑤 = dom 𝑢)
10		eqidd 2767	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝐷 = 𝐷)
11	8, 9, 10	f1eq123d 6819	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑢:dom 𝑢–1-1→𝐷))
12	11	elrab 3653	. . . . . . . . . . 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 2766	. . . . . . . . 9 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
16	3, 2, 7, 14, 15	cycpmcl 33467	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)))
17		c0ex 11218	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
18	17	tpid1 4739	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0, 1, 2}
19		fzo0to3tp 13800	. . . . . . . . . . . . . . . . 17 ⊢ (0..^3) = {0, 1, 2}
20	18, 19	eleqtrri 2865	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0..^3)
21	4	elin2d 4161	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ (^◡♯ “ {3}))
22		hashf 14394	. . . . . . . . . . . . . . . . . . . 20 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
23		ffn 6712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
24		elpreima 7060	. . . . . . . . . . . . . . . . . . . 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 4611	. . . . . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (0..^(♯‘𝑢)) = (0..^3))
30	20, 29	eleqtrrid 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 0 ∈ (0..^(♯‘𝑢)))
31		wrdsymbcl 14584	. . . . . . . . . . . . . . 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 12329	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ {0, 1, 2}
34	33, 19	eleqtrri 2865	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0..^3)
35	34, 29	eleqtrrid 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 1 ∈ (0..^(♯‘𝑢)))
36		wrdsymbcl 14584	. . . . . . . . . . . . . . 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 12336	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
39	38	tpid3 4744	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ {0, 1, 2}
40	39, 19	eleqtrri 2865	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ (0..^3)
41	40, 29	eleqtrrid 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 2 ∈ (0..^(♯‘𝑢)))
42		wrdsymbcl 14584	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) = (𝑢‘0))
46		eqidd 2767	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) = (𝑢‘1))
47		eqidd 2767	. . . . . . . . . . . . . 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 15024	. . . . . . . . . . . . . 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 6892	. . . . . . . . . . 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 14578	. . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^3))
56	55, 19	eqtrdi 2817	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = {0, 1, 2})
57		f1eq2 6777	. . . . . . . . . . . . . . . 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 11221	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
61	17, 60, 38	3pm3.2i 1358	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ V)
62		0ne1 12330	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
63		0ne2 12468	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 2
64		1ne2 12469	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 2
65	62, 63, 64	3pm3.2i 1358	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)
66		eqid 2766	. . . . . . . . . . . . . . . . 17 ⊢ {0, 1, 2} = {0, 1, 2}
67	66	f13dfv 7283	. . . . . . . . . . . . . . . 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 3016	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ≠ (𝑢‘0))
75		eqid 2766	. . . . . . . . . . . 12 ⊢ (+_g‘(SymGrp‘𝐷)) = (+_g‘(SymGrp‘𝐷))
76	3, 15, 2, 32, 37, 43, 71, 72, 74, 75	cyc3co2 33491	. . . . . . . . . . 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 33471	. . . . . . . . . . . 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 33471	. . . . . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
80	15, 79, 75	symgov 19485	. . . . . . . . . . . 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 2805	. . . . . . . . . 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 6892	. . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
85	15, 84, 79	psgnco 21770	. . . . . . . . . 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 2766	. . . . . . . . . . . . . 14 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 2, 32, 43, 73, 87	cycpm2tr 33470	. . . . . . . . . . . . 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 4793	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ⊆ 𝐷)
90		enpr2 10007	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 15 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
93	87, 92	pmtrrn 19558	. . . . . . . . . . . . . 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 2866	. . . . . . . . . . . 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 19611	. . . . . . . . . . . 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 33470	. . . . . . . . . . . . 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 4793	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ⊆ 𝐷)
100		enpr2 10007	. . . . . . . . . . . . . . 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 19558	. . . . . . . . . . . . . 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 2866	. . . . . . . . . . . 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 19611	. . . . . . . . . . . 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 7441	. . . . . . . . . 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 12474	. . . . . . . . . 10 ⊢ (-1 · -1) = 1
109	107, 108	eqtrdi 2817	. . . . . . . . 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 2805	. . . . . . . 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 21774	. . . . . . . . 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 2877	. . . . . 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 2867	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → 𝑝 ∈ 𝐴)
118		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑢(toCyc‘𝐷)
119	3, 15, 79	tocycf 33468	. . . . . . 7 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷):{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘(SymGrp‘𝐷)))
120	119	ffnd 6713	. . . . . 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 2876	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ ((toCyc‘𝐷) “ (^◡♯ “ {3})))
125	118, 121, 124	fvelimad 6955	. . . 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 3176	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐴)
127	126	ex 418	. 2 ⊢ (𝐷 ∈ Fin → (𝑝 ∈ 𝐶 → 𝑝 ∈ 𝐴))
128	127	ssrdv 3946	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 2146 ≠ wne 2961 {crab 3419 Vcvv 3458 ∪ cun 3906 ∩ cin 3907 ⊆ wss 3908 {csn 4594 {cpr 4596 {ctp 4598 class class class wbr 5114 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 “ cima 5669 ∘ ccom 5670 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7423 2_oc2o 8456 ≈ cen 8949 Fincfn 8952 0cc0 11118 1c1 11119 · cmul 11123 +∞cpnf 11258 -cneg 11460 2c2 12313 3c3 12314 ℕ₀cn0 12522 ..^cfzo 13701 ♯chash 14386 Word cword 14570 ⟨“cs2 14904 ⟨“cs3 14905 Basecbs 17294 +_gcplusg 17335 SymGrpcsymg 19470 pmTrspcpmtr 19542 pmSgncpsgn 19590 pmEvencevpm 19591 toCycctocyc 33457

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197 ax-mulf 11198

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-xnn0 12596 df-z 12610 df-dec 12730 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-hash 14387 df-word 14571 df-lsw 14620 df-concat 14628 df-s1 14655 df-substr 14701 df-pfx 14733 df-splice 14811 df-reverse 14820 df-csh 14852 df-s2 14911 df-s3 14912 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-0g 17519 df-gsum 17520 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-efmnd 18959 df-grp 19034 df-minusg 19035 df-subg 19220 df-ghm 19315 df-gim 19360 df-oppg 19447 df-symg 19471 df-pmtr 19543 df-psgn 19592 df-evpm 19593 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-cring 20349 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-dvr 20516 df-drng 20866 df-cnfld 21560 df-tocyc 33458

This theorem is used by: cyc3genpm 33503

W3C validator