Theorem cycpmco2f1 33212

Description: The word U used in cycpmco2 33221 is injective, so it can represent a cycle and form a cyclic permutation (𝑀‘𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.)

Hypotheses

Ref	Expression
cycpmco2.c	⊢ 𝑀 = (toCyc‘𝐷)
cycpmco2.s	⊢ 𝑆 = (SymGrp‘𝐷)
cycpmco2.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
cycpmco2.w	⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
cycpmco2.i	⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
cycpmco2.j	⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
cycpmco2.e	⊢ 𝐸 = ((◡𝑊‘𝐽) + 1)
cycpmco2.1	⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)

Assertion

Ref	Expression
cycpmco2f1	⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)

Proof of Theorem cycpmco2f1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmco2.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
2		ssrab2 4018	. . . . . 6 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
3		cycpmco2.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
4		cycpmco2.c	. . . . . . . . . 10 ⊢ 𝑀 = (toCyc‘𝐷)
5		cycpmco2.s	. . . . . . . . . 10 ⊢ 𝑆 = (SymGrp‘𝐷)
6		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	4, 5, 6	tocycf 33205	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
8	1, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
9	8	fdmd 6672	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	3, 9	eleqtrd 2842	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11	2, 10	sselid 3920	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
12		pfxcl 14638	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
14		cycpmco2.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
15	14	eldifad 3902	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
16	15	s1cld 14564	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17		ccatcl 14534	. . . 4 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
18	13, 16, 17	syl2anc 590	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
19		swrdcl 14606	. . . 4 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
20	11, 19	syl 17	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
21		id 22	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
22		dmeq 5852	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23		eqidd 2741	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
24	21, 22, 23	f1eq123d 6766	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
25	24	elrab 3636	. . . . . . 7 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
26	10, 25	sylib 219	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
27	26	simprd 496	. . . . 5 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
28		cycpmco2.e	. . . . . 6 ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1)
29		f1cnv 6798	. . . . . . . . . 10 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
30		f1of 6774	. . . . . . . . . 10 ⊢ (◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ◡𝑊:ran 𝑊⟶dom 𝑊)
31	27, 29, 30	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ◡𝑊:ran 𝑊⟶dom 𝑊)
32		cycpmco2.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
33	31, 32	ffvelcdmd 7033	. . . . . . . 8 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ dom 𝑊)
34		wrddm 14481	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
35	11, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
36	33, 35	eleqtrd 2842	. . . . . . 7 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
37		fzofzp1 13717	. . . . . . 7 ⊢ ((◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
39	28, 38	eqeltrid 2844	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
40	11, 27, 39	pfxf1 33028	. . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1→𝐷)
41	15	s1f1 33029	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
42		s1rn 14560	. . . . . . 7 ⊢ (𝐼 ∈ 𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
43	15, 42	syl 17	. . . . . 6 ⊢ (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
44	43	ineq2d 4156	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45		pfxrn2 33026	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
46	11, 39, 45	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
47	46	ssrind 4179	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
48	14	eldifbd 3903	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49		disjsn 4650	. . . . . . . 8 ⊢ ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
50	48, 49	sylibr 235	. . . . . . 7 ⊢ (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
51	47, 50	sseqtrd 3958	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52		ss0 4337	. . . . . 6 ⊢ ((ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
53	51, 52	syl 17	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
54	44, 53	eqtrd 2775	. . . 4 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
55	1, 13, 16, 40, 41, 54	ccatf1 33035	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1→𝐷)
56		lencl 14493	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57		nn0fz0 13577	. . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
58	57	biimpi 217	. . . . 5 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
59	11, 56, 58	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
60	11, 39, 59, 27	swrdf1 33042	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1→𝐷)
61		ccatrn 14550	. . . . . . 7 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
62	13, 16, 61	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
63	62	ineq1d 4155	. . . . 5 ⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
64		indir 4221	. . . . 5 ⊢ ((ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
65	63, 64	eqtrdi 2791	. . . 4 ⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
66		fz0ssnn0 13574	. . . . . . . . . 10 ⊢ (0...(♯‘𝑊)) ⊆ ℕ0
67	66, 39	sselid 3920	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
68		pfxval 14634	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ ℕ0) → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
69	11, 67, 68	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
70	69	rneqd 5887	. . . . . . 7 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
71	70	ineq1d 4155	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72		0elfz 13576	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
73	67, 72	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝐸))
74		elfzuz3 13473	. . . . . . . 8 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝐸))
75		eluzfz1 13483	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
76	39, 74, 75	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝐸...(♯‘𝑊)))
77		eluzfz2 13484	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐸) → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
78	39, 74, 77	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
79	11, 73, 39, 27, 76, 78	swrdrndisj 33043	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
80	71, 79	eqtrd 2775	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81		incom 4145	. . . . . 6 ⊢ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
82	43	ineq2d 4156	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83		swrdrn2 33040	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
84	11, 39, 59, 83	syl3anc 1379	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
85	84	ssrind 4179	. . . . . . . . 9 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
86	85, 50	sseqtrd 3958	. . . . . . . 8 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87		ss0 4337	. . . . . . . 8 ⊢ ((ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) = ∅)
88	86, 87	syl 17	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) = ∅)
89	82, 88	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
90	81, 89	eqtrid 2787	. . . . 5 ⊢ (𝜑 → (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
91	80, 90	uneq12d 4106	. . . 4 ⊢ (𝜑 → ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = (∅ ∪ ∅))
92		unidm 4094	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
93	92	a1i 11	. . . 4 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
94	65, 91, 93	3eqtrd 2779	. . 3 ⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
95	1, 18, 20, 55, 60, 94	ccatf1 33035	. 2 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)):dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))–1-1→𝐷)
96		cycpmco2.1	. . . 4 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
97		ovexd 7398	. . . . . 6 ⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ V)
98	28, 97	eqeltrid 2844	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ V)
99		splval 14711	. . . . 5 ⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
100	3, 98, 98, 16, 99	syl13anc 1380	. . . 4 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
101	96, 100	eqtrid 2787	. . 3 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102	101	dmeqd 5854	. . 3 ⊢ (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103		eqidd 2741	. . 3 ⊢ (𝜑 → 𝐷 = 𝐷)
104	101, 102, 103	f1eq123d 6766	. 2 ⊢ (𝜑 → (𝑈:dom 𝑈–1-1→𝐷 ↔ (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)):dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))–1-1→𝐷))
105	95, 104	mpbird 258	1 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562  ⟨cop 4568  ⟨cotp 4570  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  ⟶wf 6488  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  0cc0 11036  1c1 11037   + caddc 11039  ℕ₀cn0 12435  ℤ_≥cuz 12786  ...cfz 13459  ..^cfzo 13606  ♯chash 14290  Word cword 14473   ++ cconcat 14530  ⟨“cs1 14556   substr csubstr 14601   prefix cpfx 14631   splice csplice 14709  Basecbs 17177  SymGrpcsymg 19342  toCycctocyc 33194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-inf 9353  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-fz 13460  df-fzo 13607  df-fl 13749  df-mod 13827  df-hash 14291  df-word 14474  df-concat 14531  df-s1 14557  df-substr 14602  df-pfx 14632  df-splice 14710  df-csh 14749  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-tset 17237  df-efmnd 18835  df-symg 19343  df-tocyc 33195

This theorem is referenced by:  cycpmco2lem4  33217  cycpmco2lem5  33218  cycpmco2lem6  33219  cycpmco2lem7  33220  cycpmco2  33221