cycpmco2lem5 - Mathbox for Thierry Arnoux

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Theorem cycpmco2lem5 32330

Description: Lemma for cycpmco2 32333. (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”⟩⟩)
cycpmco2lem.1	⊢ (𝜑 → 𝐾 ∈ ran 𝑊)
cycpmco2lem5.1	⊢ (𝜑 → (^◡𝑈‘𝐾) = ((♯‘𝑈) − 1))

**Assertion**
Ref	Expression
cycpmco2lem5	⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))

Proof of Theorem cycpmco2lem5 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmco2lem.1	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ran 𝑊)
2	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → 𝐾 ∈ ran 𝑊)
3		cycpmco2.1	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
4		cycpmco2.w	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
5		cycpmco2.e	. . . . . . . . . . . . . . . . 17 ⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
6		ovexd 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
7	5, 6	eqeltrid 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ V)
8		cycpmco2.i	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
9	8	eldifad 3960	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
10	9	s1cld 14555	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
11		splval 14703	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
12	4, 7, 7, 10, 11	syl13anc 1372	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
13	3, 12	eqtrid 2784	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
14	13	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
15		ssrab2 4077	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
16		cycpmco2.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
17		cycpmco2.c	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑀 = (toCyc‘𝐷)
18		cycpmco2.s	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑆 = (SymGrp‘𝐷)
19		eqid 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑆) = (Base‘𝑆)
20	17, 18, 19	tocycf 32317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
21	16, 20	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
22	21	fdmd 6728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
23	4, 22	eleqtrd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
24	15, 23	sselid 3980	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
25		pfxcl 14629	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
26	24, 25	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
27		ccatcl 14526	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
28	26, 10, 27	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
29		swrdcl 14597	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
30	24, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
31		ccatlen 14527	. . . . . . . . . . . . . 14 ⊢ ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷) → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
32	28, 30, 31	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
33		ccatws1len 14572	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
34	26, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
35		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
36		dmeq 5903	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
37		eqidd 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
38	35, 36, 37	f1eq123d 6825	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
39	38	elrab 3683	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
40	23, 39	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
41	40	simprd 496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
42		f1cnv 6857	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
44		f1of 6833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
45	43, 44	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
46		cycpmco2.j	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
47	45, 46	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
48		wrddm 14473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
49	24, 48	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
50	47, 49	eleqtrd 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
51		fzofzp1 13731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
53	5, 52	eqeltrid 2837	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
54		pfxlen 14635	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
55	24, 53, 54	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
56	55	oveq1d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
57	34, 56	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
58		lencl 14485	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
59	24, 58	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
60		nn0fz0 13601	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ∈ ℕ₀ ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
61	59, 60	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
62		swrdlen 14599	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
63	24, 53, 61, 62	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
64	57, 63	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
65	14, 32, 64	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
66		fz0ssnn0 13598	. . . . . . . . . . . . . . . . 17 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
67	66, 53	sselid 3980	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
68	67	nn0zd 12586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ ℤ)
69	68	peano2zd 12671	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ)
70	69	zcnd 12669	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 + 1) ∈ ℂ)
71	59	nn0cnd 12536	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
72	67	nn0cnd 12536	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℂ)
73	70, 71, 72	addsubassd 11593	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
74		1cnd 11211	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℂ)
75	72, 74, 71	addassd 11238	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
76	75	oveq1d 7426	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
77	65, 73, 76	3eqtr2d 2778	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
78	74, 71	addcld 11235	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
79	72, 78	pncan2d 11575	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
80	74, 71	addcomd 11418	. . . . . . . . . . 11 ⊢ (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
81	77, 79, 80	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
82		oveq1 7418	. . . . . . . . . 10 ⊢ ((♯‘𝑊) = 𝐸 → ((♯‘𝑊) + 1) = (𝐸 + 1))
83	81, 82	sylan9eq 2792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → (♯‘𝑈) = (𝐸 + 1))
84	83	oveq1d 7426	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → ((♯‘𝑈) − 1) = ((𝐸 + 1) − 1))
85	72, 74	pncand 11574	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 + 1) − 1) = 𝐸)
86	85	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → ((𝐸 + 1) − 1) = 𝐸)
87	84, 86	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → ((♯‘𝑈) − 1) = 𝐸)
88	87	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → (𝑈‘((♯‘𝑈) − 1)) = (𝑈‘𝐸))
89		cycpmco2lem5.1	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑈‘𝐾) = ((♯‘𝑈) − 1))
90	89	fveq2d 6895	. . . . . . . 8 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = (𝑈‘((♯‘𝑈) − 1)))
91	17, 18, 16, 4, 8, 46, 5, 3	cycpmco2f1 32324	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
92		f1f1orn 6844	. . . . . . . . . . 11 ⊢ (𝑈:dom 𝑈–1-1→𝐷 → 𝑈:dom 𝑈–1-1-onto→ran 𝑈)
93	91, 92	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1-onto→ran 𝑈)
94		ssun1 4172	. . . . . . . . . . . 12 ⊢ ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
95	17, 18, 16, 4, 8, 46, 5, 3	cycpmco2rn 32325	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
96	94, 95	sseqtrrid 4035	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑊 ⊆ ran 𝑈)
97	96	sselda 3982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → 𝐾 ∈ ran 𝑈)
98		f1ocnvfv2 7277	. . . . . . . . . 10 ⊢ ((𝑈:dom 𝑈–1-1-onto→ran 𝑈 ∧ 𝐾 ∈ ran 𝑈) → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
99	93, 97, 98	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
100	1, 99	mpdan 685	. . . . . . . 8 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
101	90, 100	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → (𝑈‘((♯‘𝑈) − 1)) = 𝐾)
102	101	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → (𝑈‘((♯‘𝑈) − 1)) = 𝐾)
103	17, 18, 16, 4, 8, 46, 5, 3	cycpmco2lem2 32327	. . . . . . 7 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)
104	103	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → (𝑈‘𝐸) = 𝐼)
105	88, 102, 104	3eqtr3d 2780	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → 𝐾 = 𝐼)
106	8	eldifbd 3961	. . . . . 6 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
107	106	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → ¬ 𝐼 ∈ ran 𝑊)
108	105, 107	eqneltrd 2853	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → ¬ 𝐾 ∈ ran 𝑊)
109	2, 108	pm2.21dd 194	. . 3 ⊢ ((𝜑 ∧ (♯‘𝑊) = 𝐸) → ((𝑀‘𝑈)‘(𝑈‘((♯‘𝑈) − 1))) = ((𝑀‘𝑊)‘(𝑈‘((♯‘𝑈) − 1))))
110		splcl 14704	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
111	24, 10, 110	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
112	3, 111	eqeltrid 2837	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Word 𝐷)
113		nn0p1gt0 12503	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ₀ → 0 < ((♯‘𝑊) + 1))
114	59, 113	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 < ((♯‘𝑊) + 1))
115	114, 81	breqtrrd 5176	. . . . . 6 ⊢ (𝜑 → 0 < (♯‘𝑈))
116		eqidd 2733	. . . . . 6 ⊢ (𝜑 → ((♯‘𝑈) − 1) = ((♯‘𝑈) − 1))
117	17, 16, 112, 91, 115, 116	cycpmfv2 32314	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘((♯‘𝑈) − 1))) = (𝑈‘0))
118	117	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((𝑀‘𝑈)‘(𝑈‘((♯‘𝑈) − 1))) = (𝑈‘0))
119		f1f 6787	. . . . . . . . . . . 12 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊⟶𝐷)
120	41, 119	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝐷)
121	120	frnd 6725	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
122	16, 121	ssexd 5324	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑊 ∈ V)
123	46	ne0d 4335	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑊 ≠ ∅)
124		hashgt0 14350	. . . . . . . . 9 ⊢ ((ran 𝑊 ∈ V ∧ ran 𝑊 ≠ ∅) → 0 < (♯‘ran 𝑊))
125	122, 123, 124	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 0 < (♯‘ran 𝑊))
126	4	dmexd 7898	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑊 ∈ V)
127		hashf1rn 14314	. . . . . . . . 9 ⊢ ((dom 𝑊 ∈ V ∧ 𝑊:dom 𝑊–1-1→𝐷) → (♯‘𝑊) = (♯‘ran 𝑊))
128	126, 41, 127	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑊) = (♯‘ran 𝑊))
129	125, 128	breqtrrd 5176	. . . . . . 7 ⊢ (𝜑 → 0 < (♯‘𝑊))
130		eqidd 2733	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑊) − 1) = ((♯‘𝑊) − 1))
131	17, 16, 24, 41, 129, 130	cycpmfv2 32314	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑊‘((♯‘𝑊) − 1))) = (𝑊‘0))
132	131	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((𝑀‘𝑊)‘(𝑊‘((♯‘𝑊) − 1))) = (𝑊‘0))
133	3	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩))
134	17, 18, 16, 4, 8, 46, 5, 3	cycpmco2lem3 32328	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
135	72, 74	addcomd 11418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 + 1) = (1 + 𝐸))
136	135	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝜑 → ((((♯‘𝑊) − 1) − 𝐸) + (𝐸 + 1)) = ((((♯‘𝑊) − 1) − 𝐸) + (1 + 𝐸)))
137	71, 74	subcld 11573	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) − 1) ∈ ℂ)
138	137, 72, 74	nppcan3d 11600	. . . . . . . . . . 11 ⊢ (𝜑 → ((((♯‘𝑊) − 1) − 𝐸) + (1 + 𝐸)) = (((♯‘𝑊) − 1) + 1))
139	71, 74	npcand 11577	. . . . . . . . . . 11 ⊢ (𝜑 → (((♯‘𝑊) − 1) + 1) = (♯‘𝑊))
140	136, 138, 139	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → ((((♯‘𝑊) − 1) − 𝐸) + (𝐸 + 1)) = (♯‘𝑊))
141	134, 140	eqtr4d 2775	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑈) − 1) = ((((♯‘𝑊) − 1) − 𝐸) + (𝐸 + 1)))
142	141	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((♯‘𝑈) − 1) = ((((♯‘𝑊) − 1) − 𝐸) + (𝐸 + 1)))
143	133, 142	fveq12d 6898	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (𝑈‘((♯‘𝑈) − 1)) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘((((♯‘𝑊) − 1) − 𝐸) + (𝐸 + 1))))
144	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → 𝑊 ∈ Word 𝐷)
145		nn0fz0 13601	. . . . . . . . . 10 ⊢ (𝐸 ∈ ℕ₀ ↔ 𝐸 ∈ (0...𝐸))
146	67, 145	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (0...𝐸))
147	146	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → 𝐸 ∈ (0...𝐸))
148	53	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → 𝐸 ∈ (0...(♯‘𝑊)))
149	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ⟨“𝐼”⟩ ∈ Word 𝐷)
150	71	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (♯‘𝑊) ∈ ℂ)
151		1cnd 11211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → 1 ∈ ℂ)
152	72	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → 𝐸 ∈ ℂ)
153	150, 151, 152	sub32d 11605	. . . . . . . . 9 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (((♯‘𝑊) − 1) − 𝐸) = (((♯‘𝑊) − 𝐸) − 1))
154		fznn0sub 13535	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → ((♯‘𝑊) − 𝐸) ∈ ℕ₀)
155	53, 154	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) − 𝐸) ∈ ℕ₀)
156	155	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((♯‘𝑊) − 𝐸) ∈ ℕ₀)
157		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (♯‘𝑊) ≠ 𝐸)
158	150, 152, 156, 157	subne0nn 32065	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((♯‘𝑊) − 𝐸) ∈ ℕ)
159		fzo0end 13726	. . . . . . . . . 10 ⊢ (((♯‘𝑊) − 𝐸) ∈ ℕ → (((♯‘𝑊) − 𝐸) − 1) ∈ (0..^((♯‘𝑊) − 𝐸)))
160	158, 159	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (((♯‘𝑊) − 𝐸) − 1) ∈ (0..^((♯‘𝑊) − 𝐸)))
161	153, 160	eqeltrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (((♯‘𝑊) − 1) − 𝐸) ∈ (0..^((♯‘𝑊) − 𝐸)))
162		s1len 14558	. . . . . . . . . . 11 ⊢ (♯‘⟨“𝐼”⟩) = 1
163	162	eqcomi 2741	. . . . . . . . . 10 ⊢ 1 = (♯‘⟨“𝐼”⟩)
164	163	oveq2i 7422	. . . . . . . . 9 ⊢ (𝐸 + 1) = (𝐸 + (♯‘⟨“𝐼”⟩))
165	164	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (𝐸 + 1) = (𝐸 + (♯‘⟨“𝐼”⟩)))
166	144, 147, 148, 149, 161, 165	splfv3 32160	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘((((♯‘𝑊) − 1) − 𝐸) + (𝐸 + 1))) = (𝑊‘((((♯‘𝑊) − 1) − 𝐸) + 𝐸)))
167	137, 72	npcand 11577	. . . . . . . . 9 ⊢ (𝜑 → ((((♯‘𝑊) − 1) − 𝐸) + 𝐸) = ((♯‘𝑊) − 1))
168	167	fveq2d 6895	. . . . . . . 8 ⊢ (𝜑 → (𝑊‘((((♯‘𝑊) − 1) − 𝐸) + 𝐸)) = (𝑊‘((♯‘𝑊) − 1)))
169	168	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (𝑊‘((((♯‘𝑊) − 1) − 𝐸) + 𝐸)) = (𝑊‘((♯‘𝑊) − 1)))
170	143, 166, 169	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (𝑈‘((♯‘𝑈) − 1)) = (𝑊‘((♯‘𝑊) − 1)))
171	170	fveq2d 6895	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((𝑀‘𝑊)‘(𝑈‘((♯‘𝑈) − 1))) = ((𝑀‘𝑊)‘(𝑊‘((♯‘𝑊) − 1))))
172	13	fveq1d 6893	. . . . . . 7 ⊢ (𝜑 → (𝑈‘0) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0))
173		nn0p1nn 12513	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ₀ → (𝐸 + 1) ∈ ℕ)
174	67, 173	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 + 1) ∈ ℕ)
175		lbfzo0 13674	. . . . . . . . . . 11 ⊢ (0 ∈ (0..^(𝐸 + 1)) ↔ (𝐸 + 1) ∈ ℕ)
176	174, 175	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0..^(𝐸 + 1)))
177	57	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
178	176, 177	eleqtrrd 2836	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
179		ccatval1 14529	. . . . . . . . 9 ⊢ ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷 ∧ 0 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0))
180	28, 30, 178, 179	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0))
181		elfzonn0 13679	. . . . . . . . . . . . . 14 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → (^◡𝑊‘𝐽) ∈ ℕ₀)
182	50, 181	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ ℕ₀)
183		nn0p1nn 12513	. . . . . . . . . . . . 13 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
184	182, 183	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
185	5, 184	eqeltrid 2837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℕ)
186		lbfzo0 13674	. . . . . . . . . . 11 ⊢ (0 ∈ (0..^𝐸) ↔ 𝐸 ∈ ℕ)
187	185, 186	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0..^𝐸))
188	55	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘(𝑊 prefix 𝐸))) = (0..^𝐸))
189	187, 188	eleqtrrd 2836	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^(♯‘(𝑊 prefix 𝐸))))
190		ccatval1 14529	. . . . . . . . 9 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷 ∧ 0 ∈ (0..^(♯‘(𝑊 prefix 𝐸)))) → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0) = ((𝑊 prefix 𝐸)‘0))
191	26, 10, 189, 190	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0) = ((𝑊 prefix 𝐸)‘0))
192		nn0p1gt0 12503	. . . . . . . . . . . . . 14 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → 0 < ((^◡𝑊‘𝐽) + 1))
193	182, 192	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < ((^◡𝑊‘𝐽) + 1))
194	193, 5	breqtrrdi 5190	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐸)
195	194	gt0ne0d 11780	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 0)
196		fzne1 32037	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ (0...(♯‘𝑊)) ∧ 𝐸 ≠ 0) → 𝐸 ∈ ((0 + 1)...(♯‘𝑊)))
197	53, 195, 196	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ((0 + 1)...(♯‘𝑊)))
198		0p1e1 12336	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
199	198	oveq1i 7421	. . . . . . . . . 10 ⊢ ((0 + 1)...(♯‘𝑊)) = (1...(♯‘𝑊))
200	197, 199	eleqtrdi 2843	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (1...(♯‘𝑊)))
201		pfxfv0 14644	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝐸)‘0) = (𝑊‘0))
202	24, 200, 201	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝑊 prefix 𝐸)‘0) = (𝑊‘0))
203	180, 191, 202	3eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0) = (𝑊‘0))
204	172, 203	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝑈‘0) = (𝑊‘0))
205	204	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (𝑈‘0) = (𝑊‘0))
206	132, 171, 205	3eqtr4rd 2783	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → (𝑈‘0) = ((𝑀‘𝑊)‘(𝑈‘((♯‘𝑈) − 1))))
207	118, 206	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ (♯‘𝑊) ≠ 𝐸) → ((𝑀‘𝑈)‘(𝑈‘((♯‘𝑈) − 1))) = ((𝑀‘𝑊)‘(𝑈‘((♯‘𝑈) − 1))))
208	109, 207	pm2.61dane 3029	. 2 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘((♯‘𝑈) − 1))) = ((𝑀‘𝑊)‘(𝑈‘((♯‘𝑈) − 1))))
209	101	fveq2d 6895	. 2 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘((♯‘𝑈) − 1))) = ((𝑀‘𝑈)‘𝐾))
210	101	fveq2d 6895	. 2 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑈‘((♯‘𝑈) − 1))) = ((𝑀‘𝑊)‘𝐾))
211	208, 209, 210	3eqtr3d 2780	1 ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 ∅c0 4322 {csn 4628 ⟨cop 4634 ⟨cotp 4636 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 < clt 11250 − cmin 11446 ℕcn 12214 ℕ₀cn0 12474 ...cfz 13486 ..^cfzo 13629 ♯chash 14292 Word cword 14466 ++ cconcat 14522 ⟨“cs1 14547 substr csubstr 14592 prefix cpfx 14622 splice csplice 14701 Basecbs 17146 SymGrpcsymg 19236 toCycctocyc 32306

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-hash 14293 df-word 14467 df-concat 14523 df-s1 14548 df-substr 14593 df-pfx 14623 df-splice 14702 df-csh 14741 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-tset 17218 df-efmnd 18752 df-symg 19237 df-tocyc 32307

This theorem is referenced by: cycpmco2 32333

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