cycpmco2 - Mathbox for Thierry Arnoux

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Theorem cycpmco2 30825

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 2798	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
5	2, 3, 4	tocycf 30809	. . . . . . 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 6497	. . . . . . 7 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
9	7, 8	eleqtrd 2892	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
10	6, 9	ffvelrnd 6829	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑊) ∈ (Base‘𝑆))
11	3, 4	symgbasf 18496	. . . . 5 ⊢ ((𝑀‘𝑊) ∈ (Base‘𝑆) → (𝑀‘𝑊):𝐷⟶𝐷)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → (𝑀‘𝑊):𝐷⟶𝐷)
13	12	ffnd 6488	. . 3 ⊢ (𝜑 → (𝑀‘𝑊) Fn 𝐷)
14		cycpmco2.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
15	14	eldifad 3893	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
16		ssrab2 4007	. . . . . . . . . . 11 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
17	16, 9	sseldi 3913	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
18		id 22	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
19		dmeq 5736	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
20		eqidd 2799	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
21	18, 19, 20	f1eq123d 6583	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
22	21	elrab3 3629	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ 𝑊:dom 𝑊–1-1→𝐷))
23	22	biimpa 480	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
25		f1f 6549	. . . . . . . . 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 6494	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
28		cycpmco2.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
29	27, 28	sseldd 3916	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝐷)
30	14	eldifbd 3894	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
31		nelne2 3084	. . . . . . . 8 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼)
32	28, 30, 31	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → 𝐽 ≠ 𝐼)
33	32	necomd 3042	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽)
34	2, 1, 15, 29, 33, 3	cycpm2cl 30812	. . . . 5 ⊢ (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
35	3, 4	symgbasf 18496	. . . . 5 ⊢ ((𝑀‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝑀‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
36	34, 35	syl 17	. . . 4 ⊢ (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷)
37	36	ffnd 6488	. . 3 ⊢ (𝜑 → (𝑀‘⟨“𝐼𝐽”⟩) Fn 𝐷)
38	36	frnd 6494	. . 3 ⊢ (𝜑 → ran (𝑀‘⟨“𝐼𝐽”⟩) ⊆ 𝐷)
39		fnco 6437	. . 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 1368	. 2 ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
41		cycpmco2.1	. . . . . 6 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
42	15	s1cld 13948	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
43		splcl 14105	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
44	17, 42, 43	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
45	41, 44	eqeltrid 2894	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Word 𝐷)
46		cycpmco2.e	. . . . . 6 ⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
47	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2f1 30816	. . . . 5 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
48	2, 1, 45, 47, 3	cycpmcl 30808	. . . 4 ⊢ (𝜑 → (𝑀‘𝑈) ∈ (Base‘𝑆))
49	3, 4	symgbasf 18496	. . . 4 ⊢ ((𝑀‘𝑈) ∈ (Base‘𝑆) → (𝑀‘𝑈):𝐷⟶𝐷)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → (𝑀‘𝑈):𝐷⟶𝐷)
51	50	ffnd 6488	. 2 ⊢ (𝜑 → (𝑀‘𝑈) Fn 𝐷)
52		fvco3 6737	. . . 4 ⊢ (((𝑀‘⟨“𝐼𝐽”⟩):𝐷⟶𝐷 ∧ 𝑖 ∈ 𝐷) → (((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
53	36, 52	sylan 583	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)))
54	2, 1, 15, 29, 33, 3	cyc2fv2 30814	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
55	54	fveq2d 6649	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀‘𝑊)‘𝐼))
56	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2lem2 30819	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)
57		f1cnv 6613	. . . . . . . . . . . . . . . 16 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
58		f1of 6590	. . . . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
61		wrddm 13864	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
62	17, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
63	60, 62	eleqtrd 2892	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
64		lencl 13876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
65	17, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
66	65	nn0cnd 11945	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
67		1cnd 10625	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
68		ovexd 7170	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
69	46, 68	eqeltrid 2894	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 ∈ V)
70		splval 14104	. . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72	41, 71	syl5eq 2845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
73	72	fveq2d 6649	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
74		pfxcl 14030	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
75	17, 74	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
76		ccatcl 13917	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
77	75, 42, 76	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
78		swrdcl 13998	. . . . . . . . . . . . . . . . . . . 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 13918	. . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
82		ccatws1len 13965	. . . . . . . . . . . . . . . . . . . . 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 13129	. . . . . . . . . . . . . . . . . . . . . . . 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 2894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
87		pfxlen 14036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
88	17, 86, 87	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
89	88	oveq1d 7150	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
90	83, 89	eqtrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
91		nn0fz0 13000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑊) ∈ ℕ₀ ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
92	65, 91	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
93		swrdlen 14000	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
94	17, 86, 92, 93	syl3anc 1368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
95	90, 94	oveq12d 7153	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
96	73, 81, 95	3eqtrd 2837	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
97		fz0ssnn0 12997	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
98	97, 86	sseldi 3913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
99	98	nn0zd 12073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 ∈ ℤ)
100	99	peano2zd 12078	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ)
101	100	zcnd 12076	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 + 1) ∈ ℂ)
102	98	nn0cnd 11945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ ℂ)
103	101, 66, 102	addsubassd 11006	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
104	102, 67, 66	addassd 10652	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
105	104	oveq1d 7150	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
106	96, 103, 105	3eqtr2d 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
107	67, 66	addcld 10649	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
108	102, 107	pncan2d 10988	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
109	67, 66	addcomd 10831	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
110	106, 108, 109	3eqtrd 2837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
111	66, 67, 110	mvrraddd 11041	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
112	111	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
113	63, 112	eleqtrrd 2893	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^((♯‘𝑈) − 1)))
114	2, 1, 45, 47, 113	cycpmfv1 30805	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = (𝑈‘((^◡𝑊‘𝐽) + 1)))
115	46	fveq2i 6648	. . . . . . . . . . 11 ⊢ (𝑈‘𝐸) = (𝑈‘((^◡𝑊‘𝐽) + 1))
116	114, 115	eqtr4di 2851	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = (𝑈‘𝐸))
117	2, 1, 17, 24, 15, 30	cycpmfv3 30807	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀‘𝑊)‘𝐼) = 𝐼)
118	56, 116, 117	3eqtr4d 2843	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑊)‘𝐼))
119	72	fveq1d 6647	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈‘(^◡𝑊‘𝐽)) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(^◡𝑊‘𝐽)))
120		fzossfzop1 13110	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ ℕ₀ → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
121	98, 120	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0..^𝐸) ⊆ (0..^(𝐸 + 1)))
122		elfzonn0 13077	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → (^◡𝑊‘𝐽) ∈ ℕ₀)
123		fzonn0p1 13109	. . . . . . . . . . . . . . . . 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 7146	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐸) = (0..^((^◡𝑊‘𝐽) + 1))
126	124, 125	eleqtrrdi 2901	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^𝐸))
127	121, 126	sseldd 3916	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(𝐸 + 1)))
128	90	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
129	127, 128	eleqtrrd 2893	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
130		ccatval1 13921	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘(^◡𝑊‘𝐽)) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(^◡𝑊‘𝐽)))
132	88	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(♯‘(𝑊 prefix 𝐸))) = (0..^𝐸))
133	126, 132	eleqtrrd 2893	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘(𝑊 prefix 𝐸))))
134		ccatval1 13921	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘(^◡𝑊‘𝐽)) = ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)))
136	119, 131, 135	3eqtrd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈‘(^◡𝑊‘𝐽)) = ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)))
137		pfxfv 14035	. . . . . . . . . . . 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 1368	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑊 prefix 𝐸)‘(^◡𝑊‘𝐽)) = (𝑊‘(^◡𝑊‘𝐽)))
139		f1f1orn 6601	. . . . . . . . . . . . 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 7012	. . . . . . . . . . . 12 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝐽 ∈ ran 𝑊) → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
142	140, 28, 141	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
143	136, 138, 142	3eqtrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈‘(^◡𝑊‘𝐽)) = 𝐽)
144	143	fveq2d 6649	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑊‘𝐽))) = ((𝑀‘𝑈)‘𝐽))
145	55, 118, 144	3eqtr2d 2839	. . . . . . . 8 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀‘𝑈)‘𝐽))
146	145	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)) = ((𝑀‘𝑈)‘𝐽))
147		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → 𝑖 = 𝐽)
148	147	fveq2d 6649	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽))
149	148	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐽)))
150	147	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑈)‘𝐽))
151	146, 149, 150	3eqtr4d 2843	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 = 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
152	1	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝐷 ∈ 𝑉)
153	15, 29	s2cld 14224	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
154	153	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
155	15, 29, 33	s2f1 30647	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
156	155	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
157	27	sselda 3915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ∈ 𝐷)
158	157	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝑖 ∈ 𝐷)
159		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑊)
160	30	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ¬ 𝐼 ∈ ran 𝑊)
161		nelne2 3084	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝑖 ≠ 𝐼)
162	159, 160, 161	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ≠ 𝐼)
163	162	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝑖 ≠ 𝐼)
164		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → 𝑖 ≠ 𝐽)
165	163, 164	nelprd 4556	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ¬ 𝑖 ∈ {𝐼, 𝐽})
166	15, 29	s2rn 30646	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
167	166	eleq2d 2875	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ 𝑖 ∈ {𝐼, 𝐽}))
168	167	notbid 321	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
169	168	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
170	165, 169	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
171	2, 152, 154, 156, 158, 170	cycpmfv3 30807	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
172	171	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘𝑖))
173	1	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝐷 ∈ 𝑉)
174	7	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝑊 ∈ dom 𝑀)
175	14	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
176	28	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝐽 ∈ ran 𝑊)
177		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝑖 ∈ ran 𝑊)
178		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → 𝑖 ≠ 𝐽)
179		simpr 488	. . . . . . . . 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 30824	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (0..^𝐸)) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
181	1	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐷 ∈ 𝑉)
182	7	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑊 ∈ dom 𝑀)
183	14	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
184	28	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝐽 ∈ ran 𝑊)
185		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖 ∈ ran 𝑊)
186	162	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → 𝑖 ≠ 𝐼)
187		simpr 488	. . . . . . . . 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 30823	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
189	1	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝐷 ∈ 𝑉)
190	7	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝑊 ∈ dom 𝑀)
191	14	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
192	28	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝐽 ∈ ran 𝑊)
193		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → 𝑖 ∈ ran 𝑊)
194		simpr 488	. . . . . . . . 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 30822	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) ∧ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
196		f1f1orn 6601	. . . . . . . . . . . . . . . 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 4099	. . . . . . . . . . . . . . . . 17 ⊢ ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
199	2, 3, 1, 7, 14, 28, 46, 41	cycpmco2rn 30817	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
200	198, 199	sseqtrrid 3968	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran 𝑊 ⊆ ran 𝑈)
201	200	sselda 3915	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → 𝑖 ∈ ran 𝑈)
202		f1ocnvdm 7019	. . . . . . . . . . . . . . 15 ⊢ ((𝑈:dom 𝑈–1-1-onto→ran 𝑈 ∧ 𝑖 ∈ ran 𝑈) → (^◡𝑈‘𝑖) ∈ dom 𝑈)
203	197, 201, 202	syl2an2r 684	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (^◡𝑈‘𝑖) ∈ dom 𝑈)
204		wrddm 13864	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ Word 𝐷 → dom 𝑈 = (0..^(♯‘𝑈)))
205	45, 204	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
206	205	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → dom 𝑈 = (0..^(♯‘𝑈)))
207	203, 206	eleqtrd 2892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (^◡𝑈‘𝑖) ∈ (0..^(♯‘𝑈)))
208	65	nn0zd 12073	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ ℤ)
209	208	peano2zd 12078	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((♯‘𝑊) + 1) ∈ ℤ)
210	110, 209	eqeltrd 2890	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑈) ∈ ℤ)
211		fzoval 13034	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑈) ∈ ℤ → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
212	210, 211	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
213	212	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (0..^(♯‘𝑈)) = (0...((♯‘𝑈) − 1)))
214	207, 213	eleqtrd 2892	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → (^◡𝑈‘𝑖) ∈ (0...((♯‘𝑈) − 1)))
215		elfzr 13145	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1))) → (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)))
218	99	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1))) → 𝐸 ∈ ℤ)
219		fzospliti 13064	. . . . . . . . . . . . . 14 ⊢ (((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) ∧ 𝐸 ∈ ℤ) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
220	217, 218, 219	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ (^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1))) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1))))
221	220	ex 416	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((^◡𝑈‘𝑖) ∈ (0..^((♯‘𝑈) − 1)) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)))))
222	221	orim1d 963	. . . . . . . . . . 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 1085	. . . . . . . . . 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 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((^◡𝑈‘𝑖) ∈ (0..^𝐸) ∨ (^◡𝑈‘𝑖) ∈ (𝐸..^((♯‘𝑈) − 1)) ∨ (^◡𝑈‘𝑖) = ((♯‘𝑈) − 1)))
226	225	adantr 484	. . . . . . . 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 1428	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑊)‘𝑖))
228	172, 227	eqtr4d 2836	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ran 𝑊) ∧ 𝑖 ≠ 𝐽) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
229	151, 228	pm2.61dane 3074	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝑊) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
230	229	adantlr 714	. . . 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 30821	. . . . . . . 8 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))
232	231	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))
233		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
234	233	fveq2d 6649	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = ((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼))
235	234	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)))
236	233	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑈)‘𝑖) = ((𝑀‘𝑈)‘𝐼))
237	232, 235, 236	3eqtr4d 2843	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 = 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
238	1	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝐷 ∈ 𝑉)
239	17	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑊 ∈ Word 𝐷)
240	24	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑊:dom 𝑊–1-1→𝐷)
241		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ∈ (𝐷 ∖ ran 𝑊))
242	241	eldifad 3893	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ∈ 𝐷)
243	241	eldifbd 3894	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ ran 𝑊)
244	2, 238, 239, 240, 242, 243	cycpmfv3 30807	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑊)‘𝑖) = 𝑖)
245	153	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
246	155	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
247		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ≠ 𝐼)
248		eldifn 4055	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝐷 ∖ ran 𝑊) → ¬ 𝑖 ∈ ran 𝑊)
249		nelne2 3084	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ ran 𝑊) → 𝐽 ≠ 𝑖)
250	28, 248, 249	syl2an 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝐽 ≠ 𝑖)
251	250	necomd 3042	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → 𝑖 ≠ 𝐽)
252	251	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑖 ≠ 𝐽)
253	247, 252	nelprd 4556	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ {𝐼, 𝐽})
254	168	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → (¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩ ↔ ¬ 𝑖 ∈ {𝐼, 𝐽}))
255	253, 254	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ ran ⟨“𝐼𝐽”⟩)
256	2, 238, 245, 246, 242, 255	cycpmfv3 30807	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖) = 𝑖)
257	256	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑊)‘𝑖))
258	45	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑈 ∈ Word 𝐷)
259	47	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → 𝑈:dom 𝑈–1-1→𝐷)
260	199	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
261		nelsn 4565	. . . . . . . . . 10 ⊢ (𝑖 ≠ 𝐼 → ¬ 𝑖 ∈ {𝐼})
262	261	adantl 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ {𝐼})
263		nelun 30282	. . . . . . . . . 10 ⊢ (ran 𝑈 = (ran 𝑊 ∪ {𝐼}) → (¬ 𝑖 ∈ ran 𝑈 ↔ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})))
264	263	biimpar 481	. . . . . . . . 9 ⊢ ((ran 𝑈 = (ran 𝑊 ∪ {𝐼}) ∧ (¬ 𝑖 ∈ ran 𝑊 ∧ ¬ 𝑖 ∈ {𝐼})) → ¬ 𝑖 ∈ ran 𝑈)
265	260, 243, 262, 264	syl12anc 835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ¬ 𝑖 ∈ ran 𝑈)
266	2, 238, 258, 259, 242, 265	cycpmfv3 30807	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑈)‘𝑖) = 𝑖)
267	244, 257, 266	3eqtr4d 2843	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) ∧ 𝑖 ≠ 𝐼) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
268	237, 267	pm2.61dane 3074	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
269	268	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐷) ∧ 𝑖 ∈ (𝐷 ∖ ran 𝑊)) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
270		undif 4388	. . . . . . . 8 ⊢ (ran 𝑊 ⊆ 𝐷 ↔ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
271	27, 270	sylib 221	. . . . . . 7 ⊢ (𝜑 → (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
272	271	eleq2d 2875	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ 𝑖 ∈ 𝐷))
273		elun 4076	. . . . . 6 ⊢ (𝑖 ∈ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) ↔ (𝑖 ∈ ran 𝑊 ∨ 𝑖 ∈ (𝐷 ∖ ran 𝑊)))
274	272, 273	bitr3di 289	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐷 ↔ (𝑖 ∈ ran 𝑊 ∨ 𝑖 ∈ (𝐷 ∖ ran 𝑊))))
275	274	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (𝑖 ∈ ran 𝑊 ∨ 𝑖 ∈ (𝐷 ∖ ran 𝑊)))
276	230, 269, 275	mpjaodan 956	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝑖)) = ((𝑀‘𝑈)‘𝑖))
277	53, 276	eqtrd 2833	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩))‘𝑖) = ((𝑀‘𝑈)‘𝑖))
278	40, 51, 277	eqfnfvd 6782	1 ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 {csn 4525 {cpr 4527 ⟨cop 4531 ⟨cotp 4533 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 − cmin 10859 ℕ₀cn0 11885 ℤcz 11969 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 ++ cconcat 13913 ⟨“cs1 13940 substr csubstr 13993 prefix cpfx 14023 splice csplice 14102 ⟨“cs2 14194 Basecbs 16475 SymGrpcsymg 18487 toCycctocyc 30798

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-csh 14142 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-efmnd 18026 df-symg 18488 df-tocyc 30799

This theorem is referenced by: (None)

W3C validator