Theorem cycpmco2f1 33081

Description: The word U used in cycpmco2 33090 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 4043	. . . . . 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 2729	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	4, 5, 6	tocycf 33074	. . . . . . . . 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 6698	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	3, 9	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11	2, 10	sselid 3944	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
12		pfxcl 14642	. . . . 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 3926	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
16	15	s1cld 14568	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17		ccatcl 14539	. . . 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 14610	. . . 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 5867	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23		eqidd 2730	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
24	21, 22, 23	f1eq123d 6792	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
25	24	elrab 3659	. . . . . . 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 6824	. . . . . . . . . 10 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
30		f1of 6800	. . . . . . . . . 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 7057	. . . . . . . 8 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ dom 𝑊)
34		wrddm 14486	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
35	11, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
36	33, 35	eleqtrd 2830	. . . . . . 7 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
37		fzofzp1 13725	. . . . . . 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 2832	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
40	11, 27, 39	pfxf1 32863	. . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1→𝐷)
41	15	s1f1 32864	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
42		s1rn 14564	. . . . . . 7 ⊢ (𝐼 ∈ 𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
43	15, 42	syl 17	. . . . . 6 ⊢ (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
44	43	ineq2d 4183	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45		pfxrn2 32861	. . . . . . . . 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 4207	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
48	14	eldifbd 3927	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49		disjsn 4675	. . . . . . . 8 ⊢ ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
50	48, 49	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
51	47, 50	sseqtrd 3983	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52		ss0 4365	. . . . . 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 2764	. . . 4 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
55	1, 13, 16, 40, 41, 54	ccatf1 32870	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1→𝐷)
56		lencl 14498	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57		nn0fz0 13586	. . . . . 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 32878	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1→𝐷)
61		ccatrn 14554	. . . . . . 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 4182	. . . . 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 4249	. . . . 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 2780	. . . 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 13583	. . . . . . . . . 10 ⊢ (0...(♯‘𝑊)) ⊆ ℕ0
67	66, 39	sselid 3944	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
68		pfxval 14638	. . . . . . . . 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 5902	. . . . . . 7 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
71	70	ineq1d 4182	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72		0elfz 13585	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
73	67, 72	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝐸))
74		elfzuz3 13482	. . . . . . . 8 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝐸))
75		eluzfz1 13492	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
76	39, 74, 75	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝐸...(♯‘𝑊)))
77		eluzfz2 13493	. . . . . . . 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 32879	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
80	71, 79	eqtrd 2764	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81		incom 4172	. . . . . 6 ⊢ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
82	43	ineq2d 4183	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83		swrdrn2 32876	. . . . . . . . . . 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 4207	. . . . . . . . 9 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
86	85, 50	sseqtrd 3983	. . . . . . . 8 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87		ss0 4365	. . . . . . . 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 2764	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
90	81, 89	eqtrid 2776	. . . . 5 ⊢ (𝜑 → (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
91	80, 90	uneq12d 4132	. . . 4 ⊢ (𝜑 → ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = (∅ ∪ ∅))
92		unidm 4120	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
93	92	a1i 11	. . . 4 ⊢ (𝜑 → (∅ ∪ ∅) = ∅)
94	65, 91, 93	3eqtrd 2768	. . 3 ⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
95	1, 18, 20, 55, 60, 94	ccatf1 32870	. 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 7422	. . . . . 6 ⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ V)
98	28, 97	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ V)
99		splval 14716	. . . . 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 2776	. . 3 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102	101	dmeqd 5869	. . 3 ⊢ (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103		eqidd 2730	. . 3 ⊢ (𝜑 → 𝐷 = 𝐷)
104	101, 102, 103	f1eq123d 6792	. 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 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589  ⟨cop 4595  ⟨cotp 4597  ^◡ccnv 5637  dom cdm 5638  ran crn 5639  ⟶wf 6507  –1-1→wf1 6508  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069   + caddc 11071  ℕ₀cn0 12442  ℤ_≥cuz 12793  ...cfz 13468  ..^cfzo 13615  ♯chash 14295  Word cword 14478   ++ cconcat 14535  ⟨“cs1 14560   substr csubstr 14605   prefix cpfx 14635   splice csplice 14714  Basecbs 17179  SymGrpcsymg 19299  toCycctocyc 33063

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-substr 14606  df-pfx 14636  df-splice 14715  df-csh 14754  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-tset 17239  df-efmnd 18796  df-symg 19300  df-tocyc 33064

This theorem is referenced by:  cycpmco2lem4  33086  cycpmco2lem5  33087  cycpmco2lem6  33088  cycpmco2lem7  33089  cycpmco2  33090