cycpmco2lem7 - Mathbox for Thierry Arnoux

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Theorem cycpmco2lem7 32759

Description: Lemma for cycpmco2 32760. (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”⟩⟩)
cycpmco2lem7.1	⊢ (𝜑 → 𝐾 ∈ ran 𝑊)
cycpmco2lem7.2	⊢ (𝜑 → 𝐾 ≠ 𝐽)
cycpmco2lem7.3	⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (0..^𝐸))

**Assertion**
Ref	Expression
cycpmco2lem7	⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))

Proof of Theorem cycpmco2lem7 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmco2.c	. . . 4 ⊢ 𝑀 = (toCyc‘𝐷)
2		cycpmco2.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
3		cycpmco2.1	. . . . 5 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
4		ssrab2 4069	. . . . . . 7 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
5		cycpmco2.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
6		cycpmco2.s	. . . . . . . . . . 11 ⊢ 𝑆 = (SymGrp‘𝐷)
7		eqid 2724	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	1, 6, 7	tocycf 32744	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
9	2, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
10	9	fdmd 6718	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11	5, 10	eleqtrd 2827	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
12	4, 11	sselid 3972	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
13		cycpmco2.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
14	13	eldifad 3952	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
15	14	s1cld 14550	. . . . . 6 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
16		splcl 14699	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
17	12, 15, 16	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
18	3, 17	eqeltrid 2829	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Word 𝐷)
19		cycpmco2.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
20		cycpmco2.e	. . . . 5 ⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
21	1, 6, 2, 5, 13, 19, 20, 3	cycpmco2f1 32751	. . . 4 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
22		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
23		dmeq 5893	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
24		eqidd 2725	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
25	22, 23, 24	f1eq123d 6815	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
26	25	elrab 3675	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
27	11, 26	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
28	27	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
29		f1cnv 6847	. . . . . . . . . . . 12 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
30		f1of 6823	. . . . . . . . . . . 12 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
31	28, 29, 30	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
32	31, 19	ffvelcdmd 7077	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
33		wrddm 14468	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
34	12, 33	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
35	32, 34	eleqtrd 2827	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
36		fzofzp1 13726	. . . . . . . . 9 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
37	35, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
38	20, 37	eqeltrid 2829	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
39		elfzuz3 13495	. . . . . . 7 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ_≥‘𝐸))
40		fzoss2 13657	. . . . . . 7 ⊢ ((♯‘𝑊) ∈ (ℤ_≥‘𝐸) → (0..^𝐸) ⊆ (0..^(♯‘𝑊)))
41	38, 39, 40	3syl 18	. . . . . 6 ⊢ (𝜑 → (0..^𝐸) ⊆ (0..^(♯‘𝑊)))
42	1, 6, 2, 5, 13, 19, 20, 3	cycpmco2lem3 32755	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
43	42	oveq2d 7417	. . . . . 6 ⊢ (𝜑 → (0..^((♯‘𝑈) − 1)) = (0..^(♯‘𝑊)))
44	41, 43	sseqtrrd 4015	. . . . 5 ⊢ (𝜑 → (0..^𝐸) ⊆ (0..^((♯‘𝑈) − 1)))
45		cycpmco2lem7.3	. . . . 5 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (0..^𝐸))
46	44, 45	sseldd 3975	. . . 4 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (0..^((♯‘𝑈) − 1)))
47	1, 2, 18, 21, 46	cycpmfv1 32740	. . 3 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑈‘𝐾))) = (𝑈‘((^◡𝑈‘𝐾) + 1)))
48		cycpmco2lem7.1	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ran 𝑊)
49		f1f1orn 6834	. . . . . . 7 ⊢ (𝑈:dom 𝑈–1-1→𝐷 → 𝑈:dom 𝑈–1-1-onto→ran 𝑈)
50	21, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1-onto→ran 𝑈)
51		ssun1 4164	. . . . . . . 8 ⊢ ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
52	1, 6, 2, 5, 13, 19, 20, 3	cycpmco2rn 32752	. . . . . . . 8 ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
53	51, 52	sseqtrrid 4027	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ ran 𝑈)
54	53	sselda 3974	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → 𝐾 ∈ ran 𝑈)
55		f1ocnvfv2 7267	. . . . . 6 ⊢ ((𝑈:dom 𝑈–1-1-onto→ran 𝑈 ∧ 𝐾 ∈ ran 𝑈) → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
56	50, 54, 55	syl2an2r 682	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
57	56	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑈‘𝐾))) = ((𝑀‘𝑈)‘𝐾))
58	48, 57	mpdan 684	. . 3 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑈‘𝐾))) = ((𝑀‘𝑈)‘𝐾))
59		f1f1orn 6834	. . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
60	28, 59	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
61	41, 34	sseqtrrd 4015	. . . . . . . 8 ⊢ (𝜑 → (0..^𝐸) ⊆ dom 𝑊)
62	61, 45	sseldd 3975	. . . . . . 7 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ dom 𝑊)
63		f1ocnvfv1 7266	. . . . . . 7 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ (^◡𝑈‘𝐾) ∈ dom 𝑊) → (^◡𝑊‘(𝑊‘(^◡𝑈‘𝐾))) = (^◡𝑈‘𝐾))
64	60, 62, 63	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (^◡𝑊‘(𝑊‘(^◡𝑈‘𝐾))) = (^◡𝑈‘𝐾))
65	3	fveq1i 6882	. . . . . . . . 9 ⊢ (𝑈‘(^◡𝑈‘𝐾)) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘(^◡𝑈‘𝐾))
66		fz0ssnn0 13593	. . . . . . . . . . . 12 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
67	66, 38	sselid 3972	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
68		nn0fz0 13596	. . . . . . . . . . 11 ⊢ (𝐸 ∈ ℕ₀ ↔ 𝐸 ∈ (0...𝐸))
69	67, 68	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (0...𝐸))
70	12, 69, 38, 15, 45	splfv1 14702	. . . . . . . . 9 ⊢ (𝜑 → ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘(^◡𝑈‘𝐾)) = (𝑊‘(^◡𝑈‘𝐾)))
71	65, 70	eqtrid 2776	. . . . . . . 8 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = (𝑊‘(^◡𝑈‘𝐾)))
72	48, 56	mpdan 684	. . . . . . . 8 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
73	71, 72	eqtr3d 2766	. . . . . . 7 ⊢ (𝜑 → (𝑊‘(^◡𝑈‘𝐾)) = 𝐾)
74	73	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → (^◡𝑊‘(𝑊‘(^◡𝑈‘𝐾))) = (^◡𝑊‘𝐾))
75	64, 74	eqtr3d 2766	. . . . 5 ⊢ (𝜑 → (^◡𝑈‘𝐾) = (^◡𝑊‘𝐾))
76	75	oveq1d 7416	. . . 4 ⊢ (𝜑 → ((^◡𝑈‘𝐾) + 1) = ((^◡𝑊‘𝐾) + 1))
77	76	fveq2d 6885	. . 3 ⊢ (𝜑 → (𝑈‘((^◡𝑈‘𝐾) + 1)) = (𝑈‘((^◡𝑊‘𝐾) + 1)))
78	47, 58, 77	3eqtr3d 2772	. 2 ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = (𝑈‘((^◡𝑊‘𝐾) + 1)))
79	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩))
80	79	fveq1d 6883	. . . 4 ⊢ (𝜑 → (𝑈‘((^◡𝑊‘𝐾) + 1)) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘((^◡𝑊‘𝐾) + 1)))
81	38	elfzelzd 13499	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℤ)
82		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1))) → (^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1)))
83	20	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 = ((^◡𝑊‘𝐽) + 1))
84	83	oveq1d 7416	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 − 1) = (((^◡𝑊‘𝐽) + 1) − 1))
85		elfzonn0 13674	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → (^◡𝑊‘𝐽) ∈ ℕ₀)
86	35, 85	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ ℕ₀)
87	86	nn0cnd 12531	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ ℂ)
88		1cnd 11206	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
89	87, 88	pncand 11569	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((^◡𝑊‘𝐽) + 1) − 1) = (^◡𝑊‘𝐽))
90	84, 89	eqtr2d 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑊‘𝐽) = (𝐸 − 1))
91	90	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (^◡𝑊‘𝐽) = (𝐸 − 1))
92		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (^◡𝑈‘𝐾) = (𝐸 − 1))
93	75	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (^◡𝑈‘𝐾) = (^◡𝑊‘𝐾))
94	91, 92, 93	3eqtr2rd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (^◡𝑊‘𝐾) = (^◡𝑊‘𝐽))
95	94	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (𝑊‘(^◡𝑊‘𝐾)) = (𝑊‘(^◡𝑊‘𝐽)))
96		f1ocnvfv2 7267	. . . . . . . . . . . 12 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝐾 ∈ ran 𝑊) → (𝑊‘(^◡𝑊‘𝐾)) = 𝐾)
97	60, 48, 96	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊‘(^◡𝑊‘𝐾)) = 𝐾)
98	97	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (𝑊‘(^◡𝑊‘𝐾)) = 𝐾)
99		f1ocnvfv2 7267	. . . . . . . . . . . 12 ⊢ ((𝑊:dom 𝑊–1-1-onto→ran 𝑊 ∧ 𝐽 ∈ ran 𝑊) → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
100	60, 19, 99	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
101	100	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (𝑊‘(^◡𝑊‘𝐽)) = 𝐽)
102	95, 98, 101	3eqtr3d 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → 𝐾 = 𝐽)
103		cycpmco2lem7.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≠ 𝐽)
104	103	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → 𝐾 ≠ 𝐽)
105	102, 104	pm2.21ddne 3018	. . . . . . . 8 ⊢ ((𝜑 ∧ (^◡𝑈‘𝐾) = (𝐸 − 1)) → (^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1)))
106		0zd 12567	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℤ)
107		nn0p1nn 12508	. . . . . . . . . . . . . 14 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
108	86, 107	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
109	20, 108	eqeltrid 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℕ)
110		0p1e1 12331	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
111	110	fveq2i 6884	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
112		nnuz 12862	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
113	111, 112	eqtr4i 2755	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
114	109, 113	eleqtrrdi 2836	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ (ℤ_≥‘(0 + 1)))
115		fzosplitsnm1 13704	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝐸 ∈ (ℤ_≥‘(0 + 1))) → (0..^𝐸) = ((0..^(𝐸 − 1)) ∪ {(𝐸 − 1)}))
116	106, 114, 115	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (0..^𝐸) = ((0..^(𝐸 − 1)) ∪ {(𝐸 − 1)}))
117	45, 116	eleqtrd 2827	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ ((0..^(𝐸 − 1)) ∪ {(𝐸 − 1)}))
118		fvex 6894	. . . . . . . . . 10 ⊢ (^◡𝑈‘𝐾) ∈ V
119		elunsn 32219	. . . . . . . . . 10 ⊢ ((^◡𝑈‘𝐾) ∈ V → ((^◡𝑈‘𝐾) ∈ ((0..^(𝐸 − 1)) ∪ {(𝐸 − 1)}) ↔ ((^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1)) ∨ (^◡𝑈‘𝐾) = (𝐸 − 1))))
120	118, 119	ax-mp 5	. . . . . . . . 9 ⊢ ((^◡𝑈‘𝐾) ∈ ((0..^(𝐸 − 1)) ∪ {(𝐸 − 1)}) ↔ ((^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1)) ∨ (^◡𝑈‘𝐾) = (𝐸 − 1)))
121	117, 120	sylib 217	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1)) ∨ (^◡𝑈‘𝐾) = (𝐸 − 1)))
122	82, 105, 121	mpjaodan 955	. . . . . . 7 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1)))
123		elfzom1elp1fzo 13696	. . . . . . 7 ⊢ ((𝐸 ∈ ℤ ∧ (^◡𝑈‘𝐾) ∈ (0..^(𝐸 − 1))) → ((^◡𝑈‘𝐾) + 1) ∈ (0..^𝐸))
124	81, 122, 123	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((^◡𝑈‘𝐾) + 1) ∈ (0..^𝐸))
125	76, 124	eqeltrrd 2826	. . . . 5 ⊢ (𝜑 → ((^◡𝑊‘𝐾) + 1) ∈ (0..^𝐸))
126	12, 69, 38, 15, 125	splfv1 14702	. . . 4 ⊢ (𝜑 → ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘((^◡𝑊‘𝐾) + 1)) = (𝑊‘((^◡𝑊‘𝐾) + 1)))
127	80, 126	eqtrd 2764	. . 3 ⊢ (𝜑 → (𝑈‘((^◡𝑊‘𝐾) + 1)) = (𝑊‘((^◡𝑊‘𝐾) + 1)))
128		1zzd 12590	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
129	81, 128	zsubcld 12668	. . . . . . . 8 ⊢ (𝜑 → (𝐸 − 1) ∈ ℤ)
130		lencl 14480	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
131		nn0fz0 13596	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) ∈ ℕ₀ ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
132	131	biimpi 215	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
133	12, 130, 132	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
134	133	elfzelzd 13499	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℤ)
135	134, 128	zsubcld 12668	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑊) − 1) ∈ ℤ)
136	109	nnred 12224	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ)
137	134	zred 12663	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
138		1red 11212	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
139		elfzle2 13502	. . . . . . . . . 10 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → 𝐸 ≤ (♯‘𝑊))
140	38, 139	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ≤ (♯‘𝑊))
141	136, 137, 138, 140	lesub1dd 11827	. . . . . . . 8 ⊢ (𝜑 → (𝐸 − 1) ≤ ((♯‘𝑊) − 1))
142		eluz 12833	. . . . . . . . 9 ⊢ (((𝐸 − 1) ∈ ℤ ∧ ((♯‘𝑊) − 1) ∈ ℤ) → (((♯‘𝑊) − 1) ∈ (ℤ_≥‘(𝐸 − 1)) ↔ (𝐸 − 1) ≤ ((♯‘𝑊) − 1)))
143	142	biimpar 477	. . . . . . . 8 ⊢ ((((𝐸 − 1) ∈ ℤ ∧ ((♯‘𝑊) − 1) ∈ ℤ) ∧ (𝐸 − 1) ≤ ((♯‘𝑊) − 1)) → ((♯‘𝑊) − 1) ∈ (ℤ_≥‘(𝐸 − 1)))
144	129, 135, 141, 143	syl21anc 835	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑊) − 1) ∈ (ℤ_≥‘(𝐸 − 1)))
145		fzoss2 13657	. . . . . . 7 ⊢ (((♯‘𝑊) − 1) ∈ (ℤ_≥‘(𝐸 − 1)) → (0..^(𝐸 − 1)) ⊆ (0..^((♯‘𝑊) − 1)))
146	144, 145	syl 17	. . . . . 6 ⊢ (𝜑 → (0..^(𝐸 − 1)) ⊆ (0..^((♯‘𝑊) − 1)))
147	146, 122	sseldd 3975	. . . . 5 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (0..^((♯‘𝑊) − 1)))
148	75, 147	eqeltrrd 2826	. . . 4 ⊢ (𝜑 → (^◡𝑊‘𝐾) ∈ (0..^((♯‘𝑊) − 1)))
149	1, 2, 12, 28, 148	cycpmfv1 32740	. . 3 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐾))) = (𝑊‘((^◡𝑊‘𝐾) + 1)))
150	97	fveq2d 6885	. . 3 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑊‘(^◡𝑊‘𝐾))) = ((𝑀‘𝑊)‘𝐾))
151	127, 149, 150	3eqtr2rd 2771	. 2 ⊢ (𝜑 → ((𝑀‘𝑊)‘𝐾) = (𝑈‘((^◡𝑊‘𝐾) + 1)))
152	78, 151	eqtr4d 2767	1 ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 {crab 3424 Vcvv 3466 ∖ cdif 3937 ∪ cun 3938 ⊆ wss 3940 {csn 4620 ⟨cotp 4628 class class class wbr 5138 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 ⟶wf 6529 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 0cc0 11106 1c1 11107 + caddc 11109 ≤ cle 11246 − cmin 11441 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ℤ_≥cuz 12819 ...cfz 13481 ..^cfzo 13624 ♯chash 14287 Word cword 14461 ⟨“cs1 14542 splice csplice 14696 Basecbs 17143 SymGrpcsymg 19276 toCycctocyc 32733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-hash 14288 df-word 14462 df-concat 14518 df-s1 14543 df-substr 14588 df-pfx 14618 df-splice 14697 df-csh 14736 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-tset 17215 df-efmnd 18784 df-symg 19277 df-tocyc 32734

This theorem is referenced by: cycpmco2 32760

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