Theorem cycpmco2f1 33572

Description: The word U used in cycpmco2 33581 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 4031	. . . . . 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 2762	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	4, 5, 6	tocycf 33565	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
8	1, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
9	8	fdmd 6717	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	3, 9	eleqtrd 2864	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11	2, 10	sselid 3932	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
12		pfxcl 14751	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
13	11, 12	syl 18	. . . 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 14674	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17		ccatcl 14643	. . . 4 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
18	13, 16, 17	syl2anc 596	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
19		swrdcl 14717	. . . 4 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
20	11, 19	syl 18	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
21		id 23	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
22		dmeq 5891	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23		eqidd 2763	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
24	21, 22, 23	f1eq123d 6813	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
25	24	elrab 3648	. . . . . . 7 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
26	10, 25	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
27	26	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
28		cycpmco2.e	. . . . . 6 ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1)
29		f1cnv 6846	. . . . . . . . . 10 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
30		f1of 6821	. . . . . . . . . 10 ⊢ (◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ◡𝑊:ran 𝑊⟶dom 𝑊)
31	27, 29, 30	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ◡𝑊:ran 𝑊⟶dom 𝑊)
32		cycpmco2.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
33	31, 32	ffvelcdmd 7082	. . . . . . . 8 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ dom 𝑊)
34		wrddm 14590	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
35	11, 34	syl 18	. . . . . . . 8 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
36	33, 35	eleqtrd 2864	. . . . . . 7 ⊢ (𝜑 → (◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
37		fzofzp1 13824	. . . . . . 7 ⊢ ((◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
38	36, 37	syl 18	. . . . . 6 ⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
39	28, 38	eqeltrid 2866	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
40	11, 27, 39	pfxf1 33396	. . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1→𝐷)
41	15	s1f1 14680	. . . 4 ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
42		s1rn 14670	. . . . . . 7 ⊢ (𝐼 ∈ 𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
43	15, 42	syl 18	. . . . . 6 ⊢ (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
44	43	ineq2d 4169	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45		pfxrn2 33394	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
46	11, 39, 45	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
47	46	ssrind 4192	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
48	14	eldifbd 3915	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49		disjsn 4675	. . . . . . . 8 ⊢ ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
50	48, 49	sylibr 237	. . . . . . 7 ⊢ (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
51	47, 50	sseqtrd 3970	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52		ss0 4355	. . . . . 6 ⊢ ((ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
53	51, 52	syl 18	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
54	44, 53	eqtrd 2797	. . . 4 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
55	1, 13, 16, 40, 41, 54	ccatf1 14660	. . 3 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1→𝐷)
56		lencl 14602	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57		nn0fz0 13684	. . . . . 6 ⊢ ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
58	57	biimpi 219	. . . . 5 ⊢ ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
59	11, 56, 58	3syl 19	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
60	11, 39, 59, 27	swrdf1 14723	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1→𝐷)
61		ccatrn 14659	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
63	62	ineq1d 4168	. . . . 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 4235	. . . . 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 2813	. . . 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 13681	. . . . . . . . . 10 ⊢ (0...(♯‘𝑊)) ⊆ ℕ0
67	66, 39	sselid 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℕ0)
68		pfxval 14747	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ ℕ0) → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
69	11, 67, 68	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
70	69	rneqd 5926	. . . . . . 7 ⊢ (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
71	70	ineq1d 4168	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72		0elfz 13683	. . . . . . . 8 ⊢ (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
73	67, 72	syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝐸))
74		elfzuz3 13579	. . . . . . . 8 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝐸))
75		eluzfz1 13589	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
76	39, 74, 75	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝐸...(♯‘𝑊)))
77		eluzfz2 13590	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐸) → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
78	39, 74, 77	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
79	11, 73, 39, 27, 76, 78	swrdrndisj 33405	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
80	71, 79	eqtrd 2797	. . . . 5 ⊢ (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81		incom 4158	. . . . . 6 ⊢ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
82	43	ineq2d 4169	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83		swrdrn2 33404	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
84	11, 39, 59, 83	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
85	84	ssrind 4192	. . . . . . . . 9 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
86	85, 50	sseqtrd 3970	. . . . . . . 8 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87		ss0 4355	. . . . . . . 8 ⊢ ((ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) = ∅)
88	86, 87	syl 18	. . . . . . 7 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) = ∅)
89	82, 88	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
90	81, 89	eqtrid 2809	. . . . 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 2801	. . 3 ⊢ (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
95	1, 18, 20, 55, 60, 94	ccatf1 14660	. 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 7452	. . . . . 6 ⊢ (𝜑 → ((◡𝑊‘𝐽) + 1) ∈ V)
98	28, 97	eqeltrid 2866	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ V)
99		splval 14824	. . . . 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 1399	. . . 4 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
101	96, 100	eqtrid 2809	. . 3 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102	101	dmeqd 5893	. . 3 ⊢ (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103		eqidd 2763	. . 3 ⊢ (𝜑 → 𝐷 = 𝐷)
104	101, 102, 103	f1eq123d 6813	. 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 260	1 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  {csn 4587  ⟨cop 4593  ⟨cotp 4595  ^◡ccnv 5658  dom cdm 5659  ran crn 5660  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417  0cc0 11128  1c1 11129   + caddc 11131  ℕ₀cn0 12532  ℤ_≥cuz 12891  ...cfz 13565  ..^cfzo 13713  ♯chash 14398  Word cword 14582   ++ cconcat 14639  ⟨“cs1 14666   substr csubstr 14712   prefix cpfx 14744   splice csplice 14822  Basecbs 17307  SymGrpcsymg 19502  toCycctocyc 33554

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-inf 9417  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-uz 12892  df-rp 13047  df-fz 13566  df-fzo 13714  df-fl 13857  df-mod 13935  df-hash 14399  df-word 14583  df-concat 14640  df-s1 14667  df-substr 14713  df-pfx 14745  df-splice 14823  df-csh 14864  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-tset 17367  df-efmnd 18984  df-symg 19503  df-tocyc 33555

This theorem is used by:  cycpmco2lem4  33577  cycpmco2lem5  33578  cycpmco2lem6  33579  cycpmco2lem7  33580  cycpmco2  33581