cycpmco2 - Mathbox for Thierry Arnoux

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Theorem cycpmco2 30775

Description: The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-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
cycpmco2	⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀‘𝑈))

Proof of Theorem cycpmco2 Dummy variables 𝑖 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmco2.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
2		cycpmco2.c	. . . . . . . 8 ⊢ 𝑀 = (toCyc‘𝐷)
3		cycpmco2.s	. . . . . . . 8 ⊢ 𝑆 = (SymGrp‘𝐷)
4		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
5	2, 3, 4	tocycf 30759	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
6	1, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
7		cycpmco2.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
8	6	fdmd 6523	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
9	7, 8	eleqtrd 2915	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	6, 9	ffvelrnd 6852	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑊) ∈ (Base‘𝑆))
11	3, 4	symgbasf 18504	. . . . 5 ⊢ ((𝑀‘𝑊) ∈ (Base‘𝑆) → (𝑀‘𝑊):𝐷⟶𝐷)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → (𝑀‘𝑊):𝐷⟶𝐷)
13	12	ffnd 6515	. . 3 ⊢ (𝜑 → (𝑀‘𝑊) Fn 𝐷)
14		cycpmco2.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
15	14	eldifad 3948	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
16		ssrab2 4056	. . . . . . . . . . 11 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
17	16, 9	sseldi 3965	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
18		id 22	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
19		dmeq 5772	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
20		eqidd 2822	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
21	18, 19, 20	f1eq123d 6608	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
22	21	elrab3 3681	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ 𝑊:dom 𝑊–1-1→𝐷))
23	22	biimpa 479	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) → 𝑊:dom 𝑊–1-1→𝐷)
24	17, 9, 23	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
25		f1f 6575	. . . . . . . . 9 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊⟶𝐷)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝐷)
27	26	frnd 6521	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
28		cycpmco2.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
29	27, 28	sseldd 3968	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
30	14	eldifbd 3949	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
31		nelne2 3115	. . . . . . . 8 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼)
32	28, 30, 31	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → 𝐽 ≠ 𝐼)
33	32	necomd 3071	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
34	2, 1, 15, 29, 33, 3	cycpm2cl 30762	. . . . 5 ⊢ (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
35	3, 4	symgbasf 18504	. . . . 5 ⊢ ((𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝑀‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
37	36	ffnd 6515	. . 3 ⊢ (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷)
38	36	frnd 6521	. . 3 ⊢ (𝜑 → ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷)
39		fnco 6465	. . 3 ⊢ (((𝑀‘𝑊) Fn 𝐷 ∧ (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷 ∧ ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷) → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
40	13, 37, 38, 39	syl3anc 1367	. 2 ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
41		cycpmco2.1	. . . . . 6 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
42	15	s1cld 13957	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
43		splcl 14114	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
44	17, 42, 43	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
45	41, 44	eqeltrid 2917	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Word 𝐷)
46		cycpmco2.e	. . . . . 6 ⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
47	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2f1 30766	. . . . 5 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
48	2, 1, 45, 47, 3	cycpmcl 30758	. . . 4 ⊢ (𝜑 → (𝑀‘𝑈) ∈ (Base‘𝑆))
49	3, 4	symgbasf 18504	. . . 4 ⊢ ((𝑀‘𝑈) ∈ (Base‘𝑆) → (𝑀‘𝑈):𝐷⟶𝐷)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → (𝑀‘𝑈):𝐷⟶𝐷)
51	50	ffnd 6515	. 2 ⊢ (𝜑 → (𝑀‘𝑈) Fn 𝐷)
52		fvco3 6760	. . . 4 ⊢ (((𝑀‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷 ∧ 𝑖 ∈ 𝐷) → (((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
53	36, 52	sylan 582	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
54	2, 1, 15, 29, 33, 3	cyc2fv2 30764	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
55	54	fveq2d 6674	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀‘𝑊)‘𝐼))
56	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2lem2 30769	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)
57		f1cnv 6638	. . . . . . . . . . . . . . . 16 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
58		f1of 6615	. . . . . . . . . . . . . . . 16 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
59	24, 57, 58	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
60	59, 28	ffvelrnd 6852	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
61		wrddm 13869	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
62	17, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
63	60, 62	eleqtrd 2915	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
64		lencl 13883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
65	17, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
66	65	nn0cnd 11958	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
67		1cnd 10636	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
68		ovexd 7191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
69	46, 68	eqeltrid 2917	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 ∈ V)
70		splval 14113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
71	7, 69, 69, 42, 70	syl13anc 1368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72	41, 71	syl5eq 2868	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
73	72	fveq2d 6674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
74		pfxcl 14039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
75	17, 74	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
76		ccatcl 13926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
77	75, 42, 76	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
78		swrdcl 14007	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
79	17, 78	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
80		ccatlen 13927	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((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)⟩))))
81	77, 79, 80	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
82		ccatws1len 13974	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
83	17, 74, 82	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
84		fzofzp1 13135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
85	63, 84	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
86	46, 85	eqeltrid 2917	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
87		pfxlen 14045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
88	17, 86, 87	syl2anc 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
89	88	oveq1d 7171	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
90	83, 89	eqtrd 2856	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
91		nn0fz0 13006	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑊) ∈ ℕ₀ ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
92	65, 91	sylib 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
93		swrdlen 14009	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
94	17, 86, 92, 93	syl3anc 1367	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
95	90, 94	oveq12d 7174	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
96	73, 81, 95	3eqtrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
97		fz0ssnn0 13003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
98	97, 86	sseldi 3965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
99	98	nn0zd 12086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ ℤ)
100	99	peano2zd 12091	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ)
101	100	zcnd 12089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 + 1) ∈ ℂ)
102	98	nn0cnd 11958	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ ℂ)
103	101, 66, 102	addsubassd 11017	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
104	102, 67, 66	addassd 10663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
105	104	oveq1d 7171	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
106	96, 103, 105	3eqtr2d 2862	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
107	67, 66	addcld 10660	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
108	102, 107	pncan2d 10999	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
109	67, 66	addcomd 10842	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
110	106, 108, 109	3eqtrd 2860	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
111	66, 67, 110	mvrraddd 11052	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
112	111	oveq2d 7172	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
113	63, 112	eleqtrrd 2916	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^((♯‘𝑈) − 1)))
114	2, 1, 45, 47, 113	cycpmfv1 30755	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = (𝑈‘((^◡𝑊‘𝐽) + 1)))
115	46	fveq2i 6673	. . . . . . . . . . 11 ⊢ (𝑈‘𝐸) = (𝑈‘((^◡𝑊‘𝐽) + 1))
116	114, 115	syl6eqr 2874	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = (𝑈‘𝐸))
117	2, 1, 17, 24, 15, 30	cycpmfv3 30757	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀‘𝑊)‘𝐼) = 𝐼)
118	56, 116, 117	3eqtr4d 2866	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑊)‘𝐼))
119	72	fveq1d 6672	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈‘(^◡𝑊‘𝐽)) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(^◡𝑊‘𝐽)))
120		fzossfzop1 13116	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ ℕ₀ → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
121	98, 120	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
122		elfzonn0 13083	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → (^◡𝑊‘𝐽) ∈ ℕ₀)
123		fzonn0p1 13115	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → (^◡𝑊‘𝐽) ∈ (0..^((^◡𝑊‘𝐽) + 1)))
124	63, 122, 123	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^((^◡𝑊‘𝐽) + 1)))
125	46	oveq2i 7167	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐸) = (0..^((^◡𝑊‘𝐽) + 1))
126	124, 125	eleqtrrdi 2924	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^𝐸))
127	121, 126	sseldd 3968	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(𝐸 + 1)))
128	90	oveq2d 7172	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
129	127, 128	eleqtrrd 2916	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
130		ccatval1 13930	. . . . . . . . . . . . 13 ⊢ ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷 ∧ (^◡𝑊‘𝐽) ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(^◡𝑊‘𝐽)) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(^◡𝑊‘𝐽)))
131	77, 79, 129, 130	syl3anc 1367	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(^◡𝑊‘𝐽)) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(^◡𝑊‘𝐽)))
132	88	oveq2d 7172	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(♯‘(𝑊 prefix 𝐸))) = (0..^𝐸))
133	126, 132	eleqtrrd 2916	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘(𝑊 prefix 𝐸))))
134		ccatval1 13930	. . . . . . . . . . . . 13 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷 ∧ (^◡𝑊‘𝐽) ∈ (0..^(♯‘(𝑊 prefix 𝐸)))) → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(^◡𝑊‘𝐽)) = ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)))
135	75, 42, 133, 134	syl3anc 1367	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(^◡𝑊‘𝐽)) = ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)))
136	119, 131, 135	3eqtrd 2860	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈‘(^◡𝑊‘𝐽)) = ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)))
137		pfxfv 14044	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (^◡𝑊‘𝐽) ∈ (0..^𝐸)) → ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)) = (𝑊‘(^◡𝑊‘𝐽)))
138	17, 86, 126, 137	syl3anc 1367	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)) = (𝑊‘(^◡𝑊‘𝐽)))
139		f1f1orn 6626	. . . . . . . . . . . . 13 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
140	24, 139	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
141		f1ocnvfv2 7034	. . . . . . . . . . . 12 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝐽 ∈ ran 𝑊) → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
142	140, 28, 141	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
143	136, 138, 142	3eqtrd 2860	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈‘(^◡𝑊‘𝐽)) = 𝐽)
144	143	fveq2d 6674	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑈)‘𝐽))
145	55, 118, 144	3eqtr2d 2862	. . . . . . . 8 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀‘𝑈)‘𝐽))
146	145	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀‘𝑈)‘𝐽))
147		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → 𝑖 = 𝐽)
148	147	fveq2d 6674	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽))
149	148	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)))
150	147	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑈)‘𝐽))
151	146, 149, 150	3eqtr4d 2866	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
152	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝐷 ∈ 𝑉)
153	15, 29	s2cld 14233	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
154	153	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
155	15, 29, 33	s2f1 30621	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
156	155	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
157	27	sselda 3967	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ∈ 𝐷)
158	157	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝑖 ∈ 𝐷)
159		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑊)
160	30	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ¬ 𝐼 ∈ ran 𝑊)
161		nelne2 3115	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝑖 ≠ 𝐼)
162	159, 160, 161	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ≠ 𝐼)
163	162	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝑖 ≠ 𝐼)
164		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝑖 ≠ 𝐽)
165	163, 164	nelprd 4596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ¬ 𝑖 ∈ {𝐼, 𝐽})
166	15, 29	s2rn 30620	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
167	166	eleq2d 2898	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ 𝑖 ∈ {𝐼, 𝐽}))
168	167	notbid 320	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
169	168	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
170	165, 169	mpbird 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
171	2, 152, 154, 156, 158, 170	cycpmfv3 30757	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
172	171	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘𝑖))
173	1	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝐷 ∈ 𝑉)
174	7	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝑊 ∈ dom 𝑀)
175	14	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
176	28	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝐽 ∈ ran 𝑊)
177		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝑖 ∈ ran 𝑊)
178		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝑖 ≠ 𝐽)
179		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → (^◡𝑈‘𝑖) ∈ (0..^𝐸))
180	2, 3, 173, 174, 175, 176, 46, 41, 177, 178, 179	cycpmco2lem7 30774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
181	1	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐷 ∈ 𝑉)
182	7	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑊 ∈ dom 𝑀)
183	14	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
184	28	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐽 ∈ ran 𝑊)
185		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖 ∈ ran 𝑊)
186	162	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖 ≠ 𝐼)
187		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))
188	2, 3, 181, 182, 183, 184, 46, 41, 185, 186, 187	cycpmco2lem6 30773	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
189	1	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝐷 ∈ 𝑉)
190	7	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝑊 ∈ dom 𝑀)
191	14	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
192	28	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝐽 ∈ ran 𝑊)
193		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝑖 ∈ ran 𝑊)
194		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1))
195	2, 3, 189, 190, 191, 192, 46, 41, 193, 194	cycpmco2lem5 30772	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
196		f1f1orn 6626	. . . . . . . . . . . . . . . 16 ⊢ (𝑈:dom 𝑈–1-1→𝐷 → 𝑈:dom 𝑈–1-1-onto→ran 𝑈)
197	47, 196	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1-onto→ran 𝑈)
198		ssun1 4148	. . . . . . . . . . . . . . . . 17 ⊢ ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
199	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2rn 30767	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
200	198, 199	sseqtrrid 4020	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝑊 ⊆ ran 𝑈)
201	200	sselda 3967	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑈)
202		f1ocnvdm 7041	. . . . . . . . . . . . . . 15 ⊢ ((𝑈:dom 𝑈–1-1-onto→ran 𝑈 ∧ 𝑖 ∈ ran 𝑈) → (^◡𝑈‘𝑖) ∈ dom 𝑈)
203	197, 201, 202	syl2an2r 683	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (^◡𝑈‘𝑖) ∈ dom 𝑈)
204		wrddm 13869	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ Word 𝐷 → dom 𝑈 = (0..^(♯‘𝑈)))
205	45, 204	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
206	205	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → dom 𝑈 = (0..^(♯‘𝑈)))
207	203, 206	eleqtrd 2915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (^◡𝑈‘𝑖) ∈ (0..^(♯‘𝑈)))
208	65	nn0zd 12086	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ ℤ)
209	208	peano2zd 12091	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((♯‘𝑊) + 1) ∈ ℤ)
210	110, 209	eqeltrd 2913	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑈) ∈ ℤ)
211		fzoval 13040	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑈) ∈ ℤ → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
212	210, 211	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
213	212	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
214	207, 213	eleqtrd 2915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (^◡𝑈‘𝑖) ∈ (0...((♯‘𝑈) − 1)))
215		elfzr 13151	. . . . . . . . . . . 12 ⊢ ((^◡𝑈‘𝑖) ∈ (0...((♯‘𝑈) − 1)) → ((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
216	214, 215	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
217		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1))) → (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)))
218	99	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1))) → 𝐸 ∈ ℤ)
219		fzospliti 13070	. . . . . . . . . . . . . 14 ⊢ (((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∧ 𝐸 ∈ ℤ) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
220	217, 218, 219	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1))) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
221	220	ex 415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))))
222	221	orim1d 962	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → (((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1))))
223	216, 222	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
224		df-3or 1084	. . . . . . . . . 10 ⊢ (((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) ↔ (((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
225	223, 224	sylibr 236	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
226	225	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
227	180, 188, 195, 226	mpjao3dan 1427	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
228	172, 227	eqtr4d 2859	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
229	151, 228	pm2.61dane 3104	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
230	229	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐷) ∧ 𝑖 ∈ ran 𝑊) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
231	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2lem4 30771	. . . . . . . 8 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))
232	231	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))
233		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
234	233	fveq2d 6674	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼))
235	234	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)))
236	233	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑈)‘𝐼))
237	232, 235, 236	3eqtr4d 2866	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
238	1	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝐷 ∈ 𝑉)
239	17	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑊 ∈ Word 𝐷)
240	24	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑊:dom 𝑊–1-1→𝐷)
241		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ∈ (𝐷 ∖ ran 𝑊))
242	241	eldifad 3948	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ∈ 𝐷)
243	241	eldifbd 3949	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ ran 𝑊)
244	2, 238, 239, 240, 242, 243	cycpmfv3 30757	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑊)‘𝑖) = 𝑖)
245	153	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
246	155	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
247		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ≠ 𝐼)
248		eldifn 4104	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝐷 ∖ ran 𝑊) → ¬ 𝑖 ∈ ran 𝑊)
249		nelne2 3115	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ ran 𝑊) → 𝐽 ≠ 𝑖)
250	28, 248, 249	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝐽 ≠ 𝑖)
251	250	necomd 3071	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝑖 ≠ 𝐽)
252	251	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ≠ 𝐽)
253	247, 252	nelprd 4596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ {𝐼, 𝐽})
254	168	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
255	253, 254	mpbird 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
256	2, 238, 245, 246, 242, 255	cycpmfv3 30757	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
257	256	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘𝑖))
258	45	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑈 ∈ Word 𝐷)
259	47	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑈:dom 𝑈–1-1→𝐷)
260	199	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
261		nelsn 4605	. . . . . . . . . 10 ⊢ (𝑖 ≠ 𝐼 → ¬ 𝑖 ∈ {𝐼})
262	261	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ {𝐼})
263		nelun 30274	. . . . . . . . . 10 ⊢ (ran 𝑈 = (ran 𝑊 ∪ {𝐼}) → (¬ 𝑖 ∈ ran 𝑈 ↔ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})))
264	263	biimpar 480	. . . . . . . . 9 ⊢ ((ran 𝑈 = (ran 𝑊 ∪ {𝐼}) ∧ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})) → ¬ 𝑖 ∈ ran 𝑈)
265	260, 243, 262, 264	syl12anc 834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ ran 𝑈)
266	2, 238, 258, 259, 242, 265	cycpmfv3 30757	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑈)‘𝑖) = 𝑖)
267	244, 257, 266	3eqtr4d 2866	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
268	237, 267	pm2.61dane 3104	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
269	268	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐷) ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
270		elun 4125	. . . . . 6 ⊢ (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ (𝑖 ∈ ran 𝑊 ∨ 𝑖 ∈ (𝐷 ∖ ran 𝑊)))
271		undif 4430	. . . . . . . 8 ⊢ (ran 𝑊 ⊆ 𝐷 ↔ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
272	27, 271	sylib 220	. . . . . . 7 ⊢ (𝜑 → (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
273	272	eleq2d 2898	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ 𝑖 ∈ 𝐷))
274	270, 273	syl5rbbr 288	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐷 ↔ (𝑖 ∈ ran 𝑊 ∨ 𝑖 ∈ (𝐷 ∖ ran 𝑊))))
275	274	biimpa 479	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (𝑖 ∈ ran 𝑊 ∨ 𝑖 ∈ (𝐷 ∖ ran 𝑊)))
276	230, 269, 275	mpjaodan 955	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
277	53, 276	eqtrd 2856	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀‘𝑈)‘𝑖))
278	40, 51, 277	eqfnfvd 6805	1 ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 {csn 4567 {cpr 4569 ⟨cop 4573 ⟨cotp 4575 ^◡ccnv 5554 dom cdm 5555 ran crn 5556 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 − cmin 10870 ℕ₀cn0 11898 ℤcz 11982 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Word cword 13862 ++ cconcat 13922 ⟨“cs1 13949 substr csubstr 14002 prefix cpfx 14032 splice csplice 14111 ⟨“cs2 14203 Basecbs 16483 SymGrpcsymg 18495 toCycctocyc 30748

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-splice 14112 df-csh 14151 df-s2 14210 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-efmnd 18034 df-symg 18496 df-tocyc 30749

This theorem is referenced by: (None)

W3C validator