Theorem cycpmco2f1 33088

Description: The word U used in cycpmco2 33097 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 4030	. . . . . 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 2731	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	4, 5, 6	tocycf 33081	. . . . . . . . 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 6661	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	3, 9	eleqtrd 2833	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11	2, 10	sselid 3932	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
12		pfxcl 14582	. . . . 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 3914	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
16	15	s1cld 14508	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17		ccatcl 14478	. . . 4 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
18	13, 16, 17	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
19		swrdcl 14550	. . . 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 5843	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23		eqidd 2732	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
24	21, 22, 23	f1eq123d 6755	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
25	24	elrab 3647	. . . . . . 7 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
26	10, 25	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
27	26	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
28		cycpmco2.e	. . . . . 6 ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1)
29		f1cnv 6787	. . . . . . . . . 10 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
30		f1of 6763	. . . . . . . . . 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 7018	. . . . . . . 8 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ dom 𝑊)
34		wrddm 14425	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
35	11, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
36	33, 35	eleqtrd 2833	. . . . . . 7 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
37		fzofzp1 13661	. . . . . . 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 2835	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
40	11, 27, 39	pfxf1 32918	. . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1→𝐷)
41	15	s1f1 32919	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
42		s1rn 14504	. . . . . . 7 ⊢ (𝐼 ∈ 𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
43	15, 42	syl 17	. . . . . 6 ⊢ (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
44	43	ineq2d 4170	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45		pfxrn2 32916	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
46	11, 39, 45	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
47	46	ssrind 4194	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
48	14	eldifbd 3915	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49		disjsn 4664	. . . . . . . 8 ⊢ ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
50	48, 49	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
51	47, 50	sseqtrd 3971	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52		ss0 4352	. . . . . 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 2766	. . . 4 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
55	1, 13, 16, 40, 41, 54	ccatf1 32925	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1→𝐷)
56		lencl 14437	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57		nn0fz0 13522	. . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
58	57	biimpi 216	. . . . 5 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
59	11, 56, 58	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
60	11, 39, 59, 27	swrdf1 32932	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1→𝐷)
61		ccatrn 14494	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
63	62	ineq1d 4169	. . . . 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 4236	. . . . 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 2782	. . . 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 13519	. . . . . . . . . 10 ⊢ (0...(♯‘𝑊)) ⊆ ℕ0
67	66, 39	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
68		pfxval 14578	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ ℕ0) → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
69	11, 67, 68	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
70	69	rneqd 5878	. . . . . . 7 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
71	70	ineq1d 4169	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72		0elfz 13521	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
73	67, 72	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝐸))
74		elfzuz3 13418	. . . . . . . 8 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝐸))
75		eluzfz1 13428	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
76	39, 74, 75	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝐸...(♯‘𝑊)))
77		eluzfz2 13429	. . . . . . . 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 32933	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
80	71, 79	eqtrd 2766	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81		incom 4159	. . . . . 6 ⊢ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
82	43	ineq2d 4170	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83		swrdrn2 32930	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
84	11, 39, 59, 83	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
85	84	ssrind 4194	. . . . . . . . 9 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
86	85, 50	sseqtrd 3971	. . . . . . . 8 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87		ss0 4352	. . . . . . . 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 2766	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
90	81, 89	eqtrid 2778	. . . . 5 ⊢ (𝜑 → (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
91	80, 90	uneq12d 4119	. . . 4 ⊢ (𝜑 → ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = (∅ ∪ ∅))
92		unidm 4107	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
93	92	a1i 11	. . . 4 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
94	65, 91, 93	3eqtrd 2770	. . 3 ⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
95	1, 18, 20, 55, 60, 94	ccatf1 32925	. 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 7381	. . . . . 6 ⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ V)
98	28, 97	eqeltrid 2835	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ V)
99		splval 14655	. . . . 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 1374	. . . 4 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
101	96, 100	eqtrid 2778	. . 3 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102	101	dmeqd 5845	. . 3 ⊢ (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103		eqidd 2732	. . 3 ⊢ (𝜑 → 𝐷 = 𝐷)
104	101, 102, 103	f1eq123d 6755	. 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 257	1 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  {csn 4576  ⟨cop 4582  ⟨cotp 4584  ^◡ccnv 5615  dom cdm 5616  ran crn 5617  ⟶wf 6477  –1-1→wf1 6478  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  0cc0 11003  1c1 11004   + caddc 11006  ℕ₀cn0 12378  ℤ_≥cuz 12729  ...cfz 13404  ..^cfzo 13551  ♯chash 14234  Word cword 14417   ++ cconcat 14474  ⟨“cs1 14500   substr csubstr 14545   prefix cpfx 14575   splice csplice 14653  Basecbs 17117  SymGrpcsymg 19279  toCycctocyc 33070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-ot 4585  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-uz 12730  df-rp 12888  df-fz 13405  df-fzo 13552  df-fl 13693  df-mod 13771  df-hash 14235  df-word 14418  df-concat 14475  df-s1 14501  df-substr 14546  df-pfx 14576  df-splice 14654  df-csh 14693  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-tset 17177  df-efmnd 18774  df-symg 19280  df-tocyc 33071

This theorem is referenced by:  cycpmco2lem4  33093  cycpmco2lem5  33094  cycpmco2lem6  33095  cycpmco2lem7  33096  cycpmco2  33097