cyc3evpm - Mathbox for Thierry Arnoux

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Theorem cyc3evpm 33161

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 2735	. . . . . . . . 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 4179	. . . . . . . . . 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 3666	. . . . . . . . . 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 5883	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → dom 𝑤 = dom 𝑢)
10		eqidd 2736	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → 𝐷 = 𝐷)
11	8, 9, 10	f1eq123d 6810	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑢:dom 𝑢–1-1→𝐷))
12	11	elrab 3671	. . . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
16	3, 2, 7, 14, 15	cycpmcl 33127	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → ((toCyc‘𝐷)‘𝑢) ∈ (Base‘(SymGrp‘𝐷)))
17		c0ex 11229	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
18	17	tpid1 4744	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0, 1, 2}
19		fzo0to3tp 13768	. . . . . . . . . . . . . . . . 17 ⊢ (0..^3) = {0, 1, 2}
20	18, 19	eleqtrri 2833	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0..^3)
21	4	elin2d 4180	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 𝑢 ∈ (^◡♯ “ {3}))
22		hashf 14356	. . . . . . . . . . . . . . . . . . . 20 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
23		ffn 6706	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → ♯ Fn V)
24		elpreima 7048	. . . . . . . . . . . . . . . . . . . 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 4618	. . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (0..^(♯‘𝑢)) = (0..^3))
30	20, 29	eleqtrrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 0 ∈ (0..^(♯‘𝑢)))
31		wrdsymbcl 14545	. . . . . . . . . . . . . . 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 11231	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
34	33	tpid2 4746	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ {0, 1, 2}
35	34, 19	eleqtrri 2833	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0..^3)
36	35, 29	eleqtrrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 1 ∈ (0..^(♯‘𝑢)))
37		wrdsymbcl 14545	. . . . . . . . . . . . . . 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 12317	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ V
40	39	tpid3 4749	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ {0, 1, 2}
41	40, 19	eleqtrri 2833	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ (0..^3)
42	41, 29	eleqtrrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → 2 ∈ (0..^(♯‘𝑢)))
43		wrdsymbcl 14545	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘0) = (𝑢‘0))
47		eqidd 2736	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘1) = (𝑢‘1))
48		eqidd 2736	. . . . . . . . . . . . . 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 14980	. . . . . . . . . . . . . 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 6880	. . . . . . . . . . 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 14539	. . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = (0..^3))
57	56, 19	eqtrdi 2786	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → dom 𝑢 = {0, 1, 2})
58		f1eq2 6770	. . . . . . . . . . . . . . . 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 12311	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
63		0ne2 12447	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 2
64		1ne2 12448	. . . . . . . . . . . . . . . . 17 ⊢ 1 ≠ 2
65	62, 63, 64	3pm3.2i 1340	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 1 ∧ 0 ≠ 2 ∧ 1 ≠ 2)
66		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ {0, 1, 2} = {0, 1, 2}
67	66	f13dfv 7267	. . . . . . . . . . . . . . . 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 2987	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → (𝑢‘2) ≠ (𝑢‘0))
75		eqid 2735	. . . . . . . . . . . 12 ⊢ (+_g‘(SymGrp‘𝐷)) = (+_g‘(SymGrp‘𝐷))
76	3, 15, 2, 32, 38, 44, 71, 72, 74, 75	cyc3co2 33151	. . . . . . . . . . 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 33131	. . . . . . . . . . . 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 33131	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
80	15, 79, 75	symgov 19365	. . . . . . . . . . . 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 2774	. . . . . . . . . 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 6880	. . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
85	15, 84, 79	psgnco 21543	. . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
88	3, 2, 32, 44, 73, 87	cycpm2tr 33130	. . . . . . . . . . . . 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 4798	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘2)} ⊆ 𝐷)
90		enpr2 10016	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
93	87, 92	pmtrrn 19438	. . . . . . . . . . . . . 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 2834	. . . . . . . . . . . 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 19491	. . . . . . . . . . . 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 33130	. . . . . . . . . . . . 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 4798	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) → {(𝑢‘0), (𝑢‘1)} ⊆ 𝐷)
100		enpr2 10016	. . . . . . . . . . . . . . 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 19438	. . . . . . . . . . . . . 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 2834	. . . . . . . . . . . 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 19491	. . . . . . . . . . . 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 7423	. . . . . . . . . 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 12453	. . . . . . . . . 10 ⊢ (-1 · -1) = 1
109	107, 108	eqtrdi 2786	. . . . . . . . 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 2774	. . . . . . . 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 21547	. . . . . . . . 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 2845	. . . . . 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 2835	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) ∧ 𝑢 ∈ ({𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ∩ (^◡♯ “ {3}))) ∧ ((toCyc‘𝐷)‘𝑢) = 𝑝) → 𝑝 ∈ 𝐴)
118		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑢(toCyc‘𝐷)
119	3, 15, 79	tocycf 33128	. . . . . . 7 ⊢ (𝐷 ∈ Fin → (toCyc‘𝐷):{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘(SymGrp‘𝐷)))
120	119	ffnd 6707	. . . . . 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 2844	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ ((toCyc‘𝐷) “ (^◡♯ “ {3})))
125	118, 121, 124	fvelimad 6946	. . . 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 3148	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐶) → 𝑝 ∈ 𝐴)
127	126	ex 412	. 2 ⊢ (𝐷 ∈ Fin → (𝑝 ∈ 𝐶 → 𝑝 ∈ 𝐴))
128	127	ssrdv 3964	1 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 {crab 3415 Vcvv 3459 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 {csn 4601 {cpr 4603 {ctp 4605 class class class wbr 5119 ^◡ccnv 5653 dom cdm 5654 ran crn 5655 “ cima 5657 ∘ ccom 5658 Fn wfn 6526 ⟶wf 6527 –1-1→wf1 6528 ‘cfv 6531 (class class class)co 7405 2_oc2o 8474 ≈ cen 8956 Fincfn 8959 0cc0 11129 1c1 11130 · cmul 11134 +∞cpnf 11266 -cneg 11467 2c2 12295 3c3 12296 ℕ₀cn0 12501 ..^cfzo 13671 ♯chash 14348 Word cword 14531 ⟨“cs2 14860 ⟨“cs3 14861 Basecbs 17228 +_gcplusg 17271 SymGrpcsymg 19350 pmTrspcpmtr 19422 pmSgncpsgn 19470 pmEvencevpm 19471 toCycctocyc 33117

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-ot 4610 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 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 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-splice 14768 df-reverse 14777 df-csh 14807 df-s2 14867 df-s3 14868 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-0g 17455 df-gsum 17456 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-efmnd 18847 df-grp 18919 df-minusg 18920 df-subg 19106 df-ghm 19196 df-gim 19242 df-oppg 19329 df-symg 19351 df-pmtr 19423 df-psgn 19472 df-evpm 19473 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-drng 20691 df-cnfld 21316 df-tocyc 33118

This theorem is referenced by: cyc3genpm 33163

W3C validator