cycpmco2lem4 - Mathbox for Thierry Arnoux

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Theorem cycpmco2lem4 31405

Description: Lemma for cycpmco2 31409. (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
cycpmco2lem4	⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))

Proof of Theorem cycpmco2lem4 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmco2.c	. . 3 ⊢ 𝑀 = (toCyc‘𝐷)
2		cycpmco2.s	. . 3 ⊢ 𝑆 = (SymGrp‘𝐷)
3		cycpmco2.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
4		cycpmco2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
5		cycpmco2.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
6		cycpmco2.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
7		cycpmco2.e	. . 3 ⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
8		cycpmco2.1	. . 3 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
9	1, 2, 3, 4, 5, 6, 7, 8	cycpmco2lem1 31402	. 2 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑊)‘𝐽))
10	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝐷 ∈ 𝑉)
11		ssrab2 4014	. . . . . . . . 9 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
12		eqid 2739	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
13	1, 2, 12	tocycf 31393	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
14	3, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
15	14	fdmd 6620	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
16	4, 15	eleqtrd 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
17	11, 16	sselid 3920	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
18	5	eldifad 3900	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
19	18	s1cld 14317	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
20		splcl 14474	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
21	17, 19, 20	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
22	8, 21	eqeltrid 2844	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Word 𝐷)
23	22	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝑈 ∈ Word 𝐷)
24	1, 2, 3, 4, 5, 6, 7, 8	cycpmco2f1 31400	. . . . . 6 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
25	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝑈:dom 𝑈–1-1→𝐷)
26		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝐸 ∈ (0..^(♯‘𝑊)))
27	1, 2, 3, 4, 5, 6, 7, 8	cycpmco2lem3 31404	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
28	27	oveq2d 7300	. . . . . . 7 ⊢ (𝜑 → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
29	28	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
30	26, 29	eleqtrrd 2843	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝐸 ∈ (0..^((♯‘𝑈) − 1)))
31	1, 10, 23, 25, 30	cycpmfv1 31389	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑀‘𝑈)‘(𝑈‘𝐸)) = (𝑈‘(𝐸 + 1)))
32	1, 2, 3, 4, 5, 6, 7, 8	cycpmco2lem2 31403	. . . . . 6 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)
33	32	fveq2d 6787	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘𝐸)) = ((𝑀‘𝑈)‘𝐼))
34	33	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑀‘𝑈)‘(𝑈‘𝐸)) = ((𝑀‘𝑈)‘𝐼))
35	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝑊 ∈ Word 𝐷)
36		lencl 14245	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
37	17, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
38		nn0fz0 13363	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ₀ ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
39	37, 38	sylib 217	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
40	39	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
41		swrdfv0 14371	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0..^(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘0) = (𝑊‘𝐸))
42	35, 26, 40, 41	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘0) = (𝑊‘𝐸))
43		ovexd 7319	. . . . . . . . . . 11 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
44	7, 43	eqeltrid 2844	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ V)
45		splval 14473	. . . . . . . . . 10 ⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
46	4, 44, 44, 19, 45	syl13anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
47	8, 46	eqtrid 2791	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
48	47	fveq1d 6785	. . . . . . 7 ⊢ (𝜑 → (𝑈‘(𝐸 + 1)) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝐸 + 1)))
49	48	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝑈‘(𝐸 + 1)) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝐸 + 1)))
50		pfxcl 14399	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
51	17, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
52		ccatcl 14286	. . . . . . . . 9 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
53	51, 19, 52	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
54	53	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
55		swrdcl 14367	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
56	17, 55	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
57	56	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
58		1z 12359	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
59		fzoaddel 13449	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ (0..^(♯‘𝑊)) ∧ 1 ∈ ℤ) → (𝐸 + 1) ∈ ((0 + 1)..^((♯‘𝑊) + 1)))
60	58, 59	mpan2 688	. . . . . . . . . 10 ⊢ (𝐸 ∈ (0..^(♯‘𝑊)) → (𝐸 + 1) ∈ ((0 + 1)..^((♯‘𝑊) + 1)))
61		elfzolt2b 13407	. . . . . . . . . 10 ⊢ ((𝐸 + 1) ∈ ((0 + 1)..^((♯‘𝑊) + 1)) → (𝐸 + 1) ∈ ((𝐸 + 1)..^((♯‘𝑊) + 1)))
62	60, 61	syl 17	. . . . . . . . 9 ⊢ (𝐸 ∈ (0..^(♯‘𝑊)) → (𝐸 + 1) ∈ ((𝐸 + 1)..^((♯‘𝑊) + 1)))
63	62	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝐸 + 1) ∈ ((𝐸 + 1)..^((♯‘𝑊) + 1)))
64		ccatws1len 14334	. . . . . . . . . . . 12 ⊢ ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
65	51, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
66		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
67		dmeq 5815	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
68		eqidd 2740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
69	66, 67, 68	f1eq123d 6717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
70	69	elrab3 3626	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ 𝑊:dom 𝑊–1-1→𝐷))
71	70	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) → 𝑊:dom 𝑊–1-1→𝐷)
72	17, 16, 71	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
73		f1cnv 6749	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
75		f1of 6725	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
76	74, 75	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
77	76, 6	ffvelrnd 6971	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
78		wrddm 14233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
79	17, 78	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
80	77, 79	eleqtrd 2842	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
81		fzofzp1 13493	. . . . . . . . . . . . . . 15 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
82	80, 81	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
83	7, 82	eqeltrid 2844	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
84		pfxlen 14405	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
85	17, 83, 84	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
86	85	oveq1d 7299	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
87	65, 86	eqtrd 2779	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
88	87	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
89	47	fveq2d 6787	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
90		ccatlen 14287	. . . . . . . . . . . . . . 15 ⊢ ((((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)⟩))))
91	53, 56, 90	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
92		swrdlen 14369	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
93	17, 83, 39, 92	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
94	87, 93	oveq12d 7302	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
95	89, 91, 94	3eqtrd 2783	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
96		fz0ssnn0 13360	. . . . . . . . . . . . . . . . . 18 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
97	96, 83	sselid 3920	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
98	97	nn0zd 12433	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ ℤ)
99	98	peano2zd 12438	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ)
100	99	zcnd 12436	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 + 1) ∈ ℂ)
101	37	nn0cnd 12304	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
102	97	nn0cnd 12304	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℂ)
103	100, 101, 102	addsubassd 11361	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
104		1cnd 10979	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
105	102, 104, 101	addassd 11006	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
106	105	oveq1d 7299	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
107	95, 103, 106	3eqtr2d 2785	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
108	104, 101	addcld 11003	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
109	102, 108	pncan2d 11343	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
110	104, 101	addcomd 11186	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
111	107, 109, 110	3eqtrd 2783	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
112	89, 111, 91	3eqtr3rd 2788	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘𝑊) + 1))
113	112	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘𝑊) + 1))
114	88, 113	oveq12d 7302	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))..^((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))) = ((𝐸 + 1)..^((♯‘𝑊) + 1)))
115	63, 114	eleqtrrd 2843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝐸 + 1) ∈ ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))..^((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))))
116		ccatval2 14292	. . . . . . 7 ⊢ ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷 ∧ (𝐸 + 1) ∈ ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))..^((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝐸 + 1)) = ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘((𝐸 + 1) − (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))))
117	54, 57, 115, 116	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(𝐸 + 1)) = ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘((𝐸 + 1) − (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))))
118	87	oveq2d 7300	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 + 1) − (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = ((𝐸 + 1) − (𝐸 + 1)))
119	100	subidd 11329	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 + 1) − (𝐸 + 1)) = 0)
120	118, 119	eqtrd 2779	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + 1) − (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = 0)
121	120	fveq2d 6787	. . . . . . 7 ⊢ (𝜑 → ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘((𝐸 + 1) − (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))) = ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘0))
122	121	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘((𝐸 + 1) − (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))) = ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘0))
123	49, 117, 122	3eqtrd 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝑈‘(𝐸 + 1)) = ((𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)‘0))
124	7	fveq2i 6786	. . . . . . 7 ⊢ (𝑊‘𝐸) = (𝑊‘((^◡𝑊‘𝐽) + 1))
125	124	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝐸) = (𝑊‘((^◡𝑊‘𝐽) + 1)))
126	72	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝑊:dom 𝑊–1-1→𝐷)
127	7	oveq1i 7294	. . . . . . . . . 10 ⊢ (𝐸 − 1) = (((^◡𝑊‘𝐽) + 1) − 1)
128		elfzonn0 13441	. . . . . . . . . . . . 13 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → (^◡𝑊‘𝐽) ∈ ℕ₀)
129	80, 128	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ ℕ₀)
130	129	nn0cnd 12304	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ ℂ)
131	130, 104	pncand 11342	. . . . . . . . . 10 ⊢ (𝜑 → (((^◡𝑊‘𝐽) + 1) − 1) = (^◡𝑊‘𝐽))
132	127, 131	eqtr2id 2792	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑊‘𝐽) = (𝐸 − 1))
133	132	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (^◡𝑊‘𝐽) = (𝐸 − 1))
134		nn0p1gt0 12271	. . . . . . . . . . . . . 14 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → 0 < ((^◡𝑊‘𝐽) + 1))
135	129, 134	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < ((^◡𝑊‘𝐽) + 1))
136	135, 7	breqtrrdi 5117	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐸)
137	136	gt0ne0d 11548	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 0)
138	137	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝐸 ≠ 0)
139		fzo1fzo0n0 13447	. . . . . . . . . 10 ⊢ (𝐸 ∈ (1..^(♯‘𝑊)) ↔ (𝐸 ∈ (0..^(♯‘𝑊)) ∧ 𝐸 ≠ 0))
140	26, 138, 139	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → 𝐸 ∈ (1..^(♯‘𝑊)))
141		elfzo1elm1fzo0 13497	. . . . . . . . 9 ⊢ (𝐸 ∈ (1..^(♯‘𝑊)) → (𝐸 − 1) ∈ (0..^((♯‘𝑊) − 1)))
142	140, 141	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝐸 − 1) ∈ (0..^((♯‘𝑊) − 1)))
143	133, 142	eqeltrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (^◡𝑊‘𝐽) ∈ (0..^((♯‘𝑊) − 1)))
144	1, 10, 35, 126, 143	cycpmfv1 31389	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐽))) = (𝑊‘((^◡𝑊‘𝐽) + 1)))
145		f1f1orn 6736	. . . . . . . . . 10 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
146	72, 145	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
147		f1ocnvfv2 7158	. . . . . . . . 9 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝐽 ∈ ran 𝑊) → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
148	146, 6, 147	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
149	148	fveq2d 6787	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑊)‘𝐽))
150	149	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑊)‘𝐽))
151	125, 144, 150	3eqtr2rd 2786	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑀‘𝑊)‘𝐽) = (𝑊‘𝐸))
152	42, 123, 151	3eqtr4d 2789	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → (𝑈‘(𝐸 + 1)) = ((𝑀‘𝑊)‘𝐽))
153	31, 34, 152	3eqtr3rd 2788	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∈ (0..^(♯‘𝑊))) → ((𝑀‘𝑊)‘𝐽) = ((𝑀‘𝑈)‘𝐼))
154	149	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑊)‘𝐽))
155	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝐷 ∈ 𝑉)
156	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝑊 ∈ Word 𝐷)
157	72	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝑊:dom 𝑊–1-1→𝐷)
158	136	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 0 < 𝐸)
159		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝐸 = (♯‘𝑊))
160	158, 159	breqtrd 5101	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 0 < (♯‘𝑊))
161		oveq1 7291	. . . . . . 7 ⊢ (𝐸 = (♯‘𝑊) → (𝐸 − 1) = ((♯‘𝑊) − 1))
162	132, 161	sylan9eq 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → (^◡𝑊‘𝐽) = ((♯‘𝑊) − 1))
163	1, 155, 156, 157, 160, 162	cycpmfv2 31390	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐽))) = (𝑊‘0))
164	154, 163	eqtr3d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑊)‘𝐽) = (𝑊‘0))
165	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝑈 ∈ Word 𝐷)
166	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝑈:dom 𝑈–1-1→𝐷)
167		nn0p1gt0 12271	. . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℕ₀ → 0 < ((♯‘𝑊) + 1))
168	37, 167	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 < ((♯‘𝑊) + 1))
169	168, 111	breqtrrd 5103	. . . . . . 7 ⊢ (𝜑 → 0 < (♯‘𝑈))
170	169	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 0 < (♯‘𝑈))
171	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((♯‘𝑈) − 1) = (♯‘𝑊))
172	159, 171	eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → 𝐸 = ((♯‘𝑈) − 1))
173	1, 155, 165, 166, 170, 172	cycpmfv2 31390	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑈)‘(𝑈‘𝐸)) = (𝑈‘0))
174	47	fveq1d 6785	. . . . . 6 ⊢ (𝜑 → (𝑈‘0) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0))
175	174	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → (𝑈‘0) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0))
176		nn0p1nn 12281	. . . . . . . . . . 11 ⊢ (𝐸 ∈ ℕ₀ → (𝐸 + 1) ∈ ℕ)
177	97, 176	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 + 1) ∈ ℕ)
178		lbfzo0 13436	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(𝐸 + 1)) ↔ (𝐸 + 1) ∈ ℕ)
179	177, 178	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^(𝐸 + 1)))
180	87	oveq2d 7300	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
181	179, 180	eleqtrrd 2843	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
182		ccatval1 14290	. . . . . . . 8 ⊢ ((((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))
183	53, 56, 181, 182	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0))
184		nn0p1nn 12281	. . . . . . . . . . . 12 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
185	129, 184	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
186	7, 185	eqeltrid 2844	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ)
187		lbfzo0 13436	. . . . . . . . . 10 ⊢ (0 ∈ (0..^𝐸) ↔ 𝐸 ∈ ℕ)
188	186, 187	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0..^𝐸))
189	85	oveq2d 7300	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘(𝑊 prefix 𝐸))) = (0..^𝐸))
190	188, 189	eleqtrrd 2843	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0..^(♯‘(𝑊 prefix 𝐸))))
191		ccatval1 14290	. . . . . . . 8 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷 ∧ 0 ∈ (0..^(♯‘(𝑊 prefix 𝐸)))) → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0) = ((𝑊 prefix 𝐸)‘0))
192	51, 19, 190, 191	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘0) = ((𝑊 prefix 𝐸)‘0))
193		fzne1 31118	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (0...(♯‘𝑊)) ∧ 𝐸 ≠ 0) → 𝐸 ∈ ((0 + 1)...(♯‘𝑊)))
194	83, 137, 193	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ((0 + 1)...(♯‘𝑊)))
195		0p1e1 12104	. . . . . . . . . 10 ⊢ (0 + 1) = 1
196	195	oveq1i 7294	. . . . . . . . 9 ⊢ ((0 + 1)...(♯‘𝑊)) = (1...(♯‘𝑊))
197	194, 196	eleqtrdi 2850	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (1...(♯‘𝑊)))
198		pfxfv0 14414	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝐸)‘0) = (𝑊‘0))
199	17, 197, 198	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝑊 prefix 𝐸)‘0) = (𝑊‘0))
200	183, 192, 199	3eqtrd 2783	. . . . . 6 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0) = (𝑊‘0))
201	200	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘0) = (𝑊‘0))
202	173, 175, 201	3eqtrd 2783	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑈)‘(𝑈‘𝐸)) = (𝑊‘0))
203	33	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑈)‘(𝑈‘𝐸)) = ((𝑀‘𝑈)‘𝐼))
204	164, 202, 203	3eqtr2d 2785	. . 3 ⊢ ((𝜑 ∧ 𝐸 = (♯‘𝑊)) → ((𝑀‘𝑊)‘𝐽) = ((𝑀‘𝑈)‘𝐼))
205		elfzr 13509	. . . 4 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → (𝐸 ∈ (0..^(♯‘𝑊)) ∨ 𝐸 = (♯‘𝑊)))
206	83, 205	syl 17	. . 3 ⊢ (𝜑 → (𝐸 ∈ (0..^(♯‘𝑊)) ∨ 𝐸 = (♯‘𝑊)))
207	153, 204, 206	mpjaodan 956	. 2 ⊢ (𝜑 → ((𝑀‘𝑊)‘𝐽) = ((𝑀‘𝑈)‘𝐼))
208	9, 207	eqtrd 2779	1 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2107 ≠ wne 2944 {crab 3069 Vcvv 3433 ∖ cdif 3885 ⟨cop 4568 ⟨cotp 4570 class class class wbr 5075 ^◡ccnv 5589 dom cdm 5590 ran crn 5591 ⟶wf 6433 –1-1→wf1 6434 –1-1-onto→wf1o 6436 ‘cfv 6437 (class class class)co 7284 0cc0 10880 1c1 10881 + caddc 10883 < clt 11018 − cmin 11214 ℕcn 11982 ℕ₀cn0 12242 ℤcz 12328 ...cfz 13248 ..^cfzo 13391 ♯chash 14053 Word cword 14226 ++ cconcat 14282 ⟨“cs1 14309 substr csubstr 14362 prefix cpfx 14392 splice csplice 14471 ⟨“cs2 14563 Basecbs 16921 SymGrpcsymg 18983 toCycctocyc 31382

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-ot 4571 df-uni 4841 df-int 4881 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-er 8507 df-map 8626 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-sup 9210 df-inf 9211 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-z 12329 df-uz 12592 df-rp 12740 df-fz 13249 df-fzo 13392 df-fl 13521 df-mod 13599 df-hash 14054 df-word 14227 df-concat 14283 df-s1 14310 df-substr 14363 df-pfx 14393 df-splice 14472 df-csh 14511 df-s2 14570 df-struct 16857 df-sets 16874 df-slot 16892 df-ndx 16904 df-base 16922 df-ress 16951 df-plusg 16984 df-tset 16990 df-efmnd 18517 df-symg 18984 df-tocyc 31383

This theorem is referenced by: cycpmco2 31409

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