cycpmco2lem6 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem6		Structured version Visualization version GIF version

Theorem cycpmco2lem6 32560

Description: Lemma for cycpmco2 32562. (Contributed by Thierry Arnoux, 4-Jan-2024.)

**Hypotheses**
Ref	Expression
cycpmco2.c	⊢ 𝑀 = (toCyc‘𝐷)
cycpmco2.s	⊢ 𝑆 = (SymGrp‘𝐷)
cycpmco2.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
cycpmco2.w	⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
cycpmco2.i	⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
cycpmco2.j	⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
cycpmco2.e	⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
cycpmco2.1	⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
cycpmco2lem.1	⊢ (𝜑 → 𝐾 ∈ ran 𝑊)
cycpmco2lem6.2	⊢ (𝜑 → 𝐾 ≠ 𝐼)
cycpmco2lem6.1	⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))

**Assertion**
Ref	Expression
cycpmco2lem6	⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))

Proof of Theorem cycpmco2lem6 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 4076	. . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	1, 6, 7	tocycf 32546	. . . . . . . . . 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 6727	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
11	5, 10	eleqtrd 2833	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
12	4, 11	sselid 3979	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
13		cycpmco2.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
14	13	eldifad 3959	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
15	14	s1cld 14557	. . . . . 6 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
16		splcl 14706	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
17	12, 15, 16	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ∈ Word 𝐷)
18	3, 17	eqeltrid 2835	. . . 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 32553	. . . 4 ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
22		fz0ssnn0 13600	. . . . . . . 8 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
23		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
24		dmeq 5902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
25		eqidd 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
26	23, 24, 25	f1eq123d 6824	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
27	26	elrab 3682	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
28	11, 27	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
29	28	simprd 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
30		f1cnv 6856	. . . . . . . . . . . . 13 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
31		f1of 6832	. . . . . . . . . . . . 13 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
32	29, 30, 31	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
33	32, 19	ffvelcdmd 7086	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
34		wrddm 14475	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
35	12, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
36	33, 35	eleqtrd 2833	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
37		fzofzp1 13733	. . . . . . . . . 10 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
39	20, 38	eqeltrid 2835	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
40	22, 39	sselid 3979	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
41		nn0uz 12868	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
42	40, 41	eleqtrdi 2841	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (ℤ_≥‘0))
43		fzoss1 13663	. . . . . 6 ⊢ (𝐸 ∈ (ℤ_≥‘0) → (𝐸..^((♯‘𝑈) − 1)) ⊆ (0..^((♯‘𝑈) − 1)))
44	42, 43	syl 17	. . . . 5 ⊢ (𝜑 → (𝐸..^((♯‘𝑈) − 1)) ⊆ (0..^((♯‘𝑈) − 1)))
45		cycpmco2lem6.1	. . . . 5 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))
46	44, 45	sseldd 3982	. . . 4 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (0..^((♯‘𝑈) − 1)))
47	1, 2, 18, 21, 46	cycpmfv1 32542	. . 3 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑈‘𝐾))) = (𝑈‘((^◡𝑈‘𝐾) + 1)))
48		cycpmco2lem.1	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ran 𝑊)
49		f1f1orn 6843	. . . . . . 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 4171	. . . . . . . 8 ⊢ ran 𝑊 ⊆ (ran 𝑊 ∪ {𝐼})
52	1, 6, 2, 5, 13, 19, 20, 3	cycpmco2rn 32554	. . . . . . . 8 ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
53	51, 52	sseqtrrid 4034	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ ran 𝑈)
54	53	sselda 3981	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → 𝐾 ∈ ran 𝑈)
55		f1ocnvfv2 7277	. . . . . 6 ⊢ ((𝑈:dom 𝑈–1-1-onto→ran 𝑈 ∧ 𝐾 ∈ ran 𝑈) → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
56	50, 54, 55	syl2an2r 681	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
57	48, 56	mpdan 683	. . . 4 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = 𝐾)
58	57	fveq2d 6894	. . 3 ⊢ (𝜑 → ((𝑀‘𝑈)‘(𝑈‘(^◡𝑈‘𝐾))) = ((𝑀‘𝑈)‘𝐾))
59	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩))
60		fzossz 13656	. . . . . . . 8 ⊢ (𝐸..^((♯‘𝑈) − 1)) ⊆ ℤ
61	60, 45	sselid 3979	. . . . . . 7 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ ℤ)
62	61	zcnd 12671	. . . . . 6 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ ℂ)
63	40	nn0cnd 12538	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℂ)
64		1cnd 11213	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
65	62, 63, 64	nppcan3d 11602	. . . . 5 ⊢ (𝜑 → (((^◡𝑈‘𝐾) − 𝐸) + (1 + 𝐸)) = ((^◡𝑈‘𝐾) + 1))
66	65	eqcomd 2736	. . . 4 ⊢ (𝜑 → ((^◡𝑈‘𝐾) + 1) = (((^◡𝑈‘𝐾) − 𝐸) + (1 + 𝐸)))
67	59, 66	fveq12d 6897	. . 3 ⊢ (𝜑 → (𝑈‘((^◡𝑈‘𝐾) + 1)) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘(((^◡𝑈‘𝐾) − 𝐸) + (1 + 𝐸))))
68	47, 58, 67	3eqtr3d 2778	. 2 ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘(((^◡𝑈‘𝐾) − 𝐸) + (1 + 𝐸))))
69	62, 63	npcand 11579	. . . 4 ⊢ (𝜑 → (((^◡𝑈‘𝐾) − 𝐸) + 𝐸) = (^◡𝑈‘𝐾))
70	69	fveq2d 6894	. . 3 ⊢ (𝜑 → (𝑊‘(((^◡𝑈‘𝐾) − 𝐸) + 𝐸)) = (𝑊‘(^◡𝑈‘𝐾)))
71		nn0fz0 13603	. . . . 5 ⊢ (𝐸 ∈ ℕ₀ ↔ 𝐸 ∈ (0...𝐸))
72	40, 71	sylib 217	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (0...𝐸))
73		lencl 14487	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
74	12, 73	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
75	74	nn0cnd 12538	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑊) ∈ ℂ)
76		ovexd 7446	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
77	20, 76	eqeltrid 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ V)
78		splval 14705	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
79	5, 77, 77, 15, 78	syl13anc 1370	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
80	3, 79	eqtrid 2782	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
81	80	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑈) = (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
82		pfxcl 14631	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
83	12, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
84		ccatcl 14528	. . . . . . . . . . . . . 14 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
85	83, 15, 84	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
86		swrdcl 14599	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
87	12, 86	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
88		ccatlen 14529	. . . . . . . . . . . . 13 ⊢ ((((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)⟩))))
89	85, 87, 88	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
90		ccatws1len 14574	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
91	12, 82, 90	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
92		pfxlen 14637	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
93	12, 39, 92	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
94	93	oveq1d 7426	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
95	91, 94	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
96		nn0fz0 13603	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) ∈ ℕ₀ ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
97	74, 96	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
98		swrdlen 14601	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
99	12, 39, 97, 98	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐸))
100	95, 99	oveq12d 7429	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) + (♯‘(𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
101	81, 89, 100	3eqtrd 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
102	40	nn0zd 12588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℤ)
103	102	peano2zd 12673	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ)
104	103	zcnd 12671	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 + 1) ∈ ℂ)
105	104, 75, 63	addsubassd 11595	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + 1) + ((♯‘𝑊) − 𝐸)))
106	63, 64, 75	addassd 11240	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸 + 1) + (♯‘𝑊)) = (𝐸 + (1 + (♯‘𝑊))))
107	106	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 + 1) + (♯‘𝑊)) − 𝐸) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
108	101, 105, 107	3eqtr2d 2776	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑈) = ((𝐸 + (1 + (♯‘𝑊))) − 𝐸))
109	64, 75	addcld 11237	. . . . . . . . . . 11 ⊢ (𝜑 → (1 + (♯‘𝑊)) ∈ ℂ)
110	63, 109	pncan2d 11577	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 + (1 + (♯‘𝑊))) − 𝐸) = (1 + (♯‘𝑊)))
111	64, 75	addcomd 11420	. . . . . . . . . 10 ⊢ (𝜑 → (1 + (♯‘𝑊)) = ((♯‘𝑊) + 1))
112	108, 110, 111	3eqtrd 2774	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑈) = ((♯‘𝑊) + 1))
113	75, 64, 112	mvrraddd 11630	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
114	113	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → (𝐸..^((♯‘𝑈) − 1)) = (𝐸..^(♯‘𝑊)))
115	45, 114	eleqtrd 2833	. . . . . 6 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ (𝐸..^(♯‘𝑊)))
116		fzosubel 13695	. . . . . 6 ⊢ (((^◡𝑈‘𝐾) ∈ (𝐸..^(♯‘𝑊)) ∧ 𝐸 ∈ ℤ) → ((^◡𝑈‘𝐾) − 𝐸) ∈ ((𝐸 − 𝐸)..^((♯‘𝑊) − 𝐸)))
117	115, 102, 116	syl2anc 582	. . . . 5 ⊢ (𝜑 → ((^◡𝑈‘𝐾) − 𝐸) ∈ ((𝐸 − 𝐸)..^((♯‘𝑊) − 𝐸)))
118	63	subidd 11563	. . . . . 6 ⊢ (𝜑 → (𝐸 − 𝐸) = 0)
119	118	oveq1d 7426	. . . . 5 ⊢ (𝜑 → ((𝐸 − 𝐸)..^((♯‘𝑊) − 𝐸)) = (0..^((♯‘𝑊) − 𝐸)))
120	117, 119	eleqtrd 2833	. . . 4 ⊢ (𝜑 → ((^◡𝑈‘𝐾) − 𝐸) ∈ (0..^((♯‘𝑊) − 𝐸)))
121	64, 63	addcomd 11420	. . . . 5 ⊢ (𝜑 → (1 + 𝐸) = (𝐸 + 1))
122		s1len 14560	. . . . . 6 ⊢ (♯‘⟨“𝐼”⟩) = 1
123	122	oveq2i 7422	. . . . 5 ⊢ (𝐸 + (♯‘⟨“𝐼”⟩)) = (𝐸 + 1)
124	121, 123	eqtr4di 2788	. . . 4 ⊢ (𝜑 → (1 + 𝐸) = (𝐸 + (♯‘⟨“𝐼”⟩)))
125	12, 72, 39, 15, 120, 124	splfv3 32389	. . 3 ⊢ (𝜑 → ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘(((^◡𝑈‘𝐾) − 𝐸) + (1 + 𝐸))) = (𝑊‘(((^◡𝑈‘𝐾) − 𝐸) + 𝐸)))
126	113	oveq1d 7426	. . . . . . . 8 ⊢ (𝜑 → (((♯‘𝑈) − 1) − 1) = ((♯‘𝑊) − 1))
127	126	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → (𝐸..^(((♯‘𝑈) − 1) − 1)) = (𝐸..^((♯‘𝑊) − 1)))
128		fzoss1 13663	. . . . . . . 8 ⊢ (𝐸 ∈ (ℤ_≥‘0) → (𝐸..^((♯‘𝑊) − 1)) ⊆ (0..^((♯‘𝑊) − 1)))
129	42, 128	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐸..^((♯‘𝑊) − 1)) ⊆ (0..^((♯‘𝑊) − 1)))
130	127, 129	eqsstrd 4019	. . . . . 6 ⊢ (𝜑 → (𝐸..^(((♯‘𝑈) − 1) − 1)) ⊆ (0..^((♯‘𝑊) − 1)))
131		f1ocnvdm 7285	. . . . . . . . . 10 ⊢ ((𝑈:dom 𝑈–1-1-onto→ran 𝑈 ∧ 𝐾 ∈ ran 𝑈) → (^◡𝑈‘𝐾) ∈ dom 𝑈)
132	50, 54, 131	syl2an2r 681	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ ran 𝑊) → (^◡𝑈‘𝐾) ∈ dom 𝑈)
133	48, 132	mpdan 683	. . . . . . . 8 ⊢ (𝜑 → (^◡𝑈‘𝐾) ∈ dom 𝑈)
134	74	nn0zd 12588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ ℤ)
135	134	peano2zd 12673	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝑊) + 1) ∈ ℤ)
136		elfzonn0 13681	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → (^◡𝑊‘𝐽) ∈ ℕ₀)
137		nn0p1nn 12515	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑊‘𝐽) ∈ ℕ₀ → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
138	36, 136, 137	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ ℕ)
139	20, 138	eqeltrid 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ ℕ)
140	139	nnred 12231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ)
141	134	zred 12670	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
142		1red 11219	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
143		elfzle2 13509	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ (0...(♯‘𝑊)) → 𝐸 ≤ (♯‘𝑊))
144	39, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ≤ (♯‘𝑊))
145	140, 141, 142, 144	leadd1dd 11832	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 + 1) ≤ ((♯‘𝑊) + 1))
146		eluz2 12832	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) + 1) ∈ (ℤ_≥‘(𝐸 + 1)) ↔ ((𝐸 + 1) ∈ ℤ ∧ ((♯‘𝑊) + 1) ∈ ℤ ∧ (𝐸 + 1) ≤ ((♯‘𝑊) + 1)))
147	103, 135, 145, 146	syl3anbrc 1341	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) + 1) ∈ (ℤ_≥‘(𝐸 + 1)))
148		fzoss2 13664	. . . . . . . . . . . 12 ⊢ (((♯‘𝑊) + 1) ∈ (ℤ_≥‘(𝐸 + 1)) → (0..^(𝐸 + 1)) ⊆ (0..^((♯‘𝑊) + 1)))
149	147, 148	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^(𝐸 + 1)) ⊆ (0..^((♯‘𝑊) + 1)))
150		fzonn0p1 13713	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℕ₀ → 𝐸 ∈ (0..^(𝐸 + 1)))
151	40, 150	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ (0..^(𝐸 + 1)))
152	149, 151	sseldd 3982	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (0..^((♯‘𝑊) + 1)))
153	112	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘𝑈)) = (0..^((♯‘𝑊) + 1)))
154	152, 153	eleqtrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (0..^(♯‘𝑈)))
155		wrddm 14475	. . . . . . . . . 10 ⊢ (𝑈 ∈ Word 𝐷 → dom 𝑈 = (0..^(♯‘𝑈)))
156	18, 155	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
157	154, 156	eleqtrrd 2834	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ dom 𝑈)
158		cycpmco2lem6.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≠ 𝐼)
159	1, 6, 2, 5, 13, 19, 20, 3	cycpmco2lem2 32556	. . . . . . . . 9 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)
160	158, 57, 159	3netr4d 3016	. . . . . . . 8 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) ≠ (𝑈‘𝐸))
161		f1fveq 7263	. . . . . . . . . 10 ⊢ ((𝑈:dom 𝑈–1-1→𝐷 ∧ ((^◡𝑈‘𝐾) ∈ dom 𝑈 ∧ 𝐸 ∈ dom 𝑈)) → ((𝑈‘(^◡𝑈‘𝐾)) = (𝑈‘𝐸) ↔ (^◡𝑈‘𝐾) = 𝐸))
162	161	necon3bid 2983	. . . . . . . . 9 ⊢ ((𝑈:dom 𝑈–1-1→𝐷 ∧ ((^◡𝑈‘𝐾) ∈ dom 𝑈 ∧ 𝐸 ∈ dom 𝑈)) → ((𝑈‘(^◡𝑈‘𝐾)) ≠ (𝑈‘𝐸) ↔ (^◡𝑈‘𝐾) ≠ 𝐸))
163	162	biimp3a 1467	. . . . . . . 8 ⊢ ((𝑈:dom 𝑈–1-1→𝐷 ∧ ((^◡𝑈‘𝐾) ∈ dom 𝑈 ∧ 𝐸 ∈ dom 𝑈) ∧ (𝑈‘(^◡𝑈‘𝐾)) ≠ (𝑈‘𝐸)) → (^◡𝑈‘𝐾) ≠ 𝐸)
164	21, 133, 157, 160, 163	syl121anc 1373	. . . . . . 7 ⊢ (𝜑 → (^◡𝑈‘𝐾) ≠ 𝐸)
165		fzom1ne1 32279	. . . . . . 7 ⊢ (((^◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)) ∧ (^◡𝑈‘𝐾) ≠ 𝐸) → ((^◡𝑈‘𝐾) − 1) ∈ (𝐸..^(((♯‘𝑈) − 1) − 1)))
166	45, 164, 165	syl2anc 582	. . . . . 6 ⊢ (𝜑 → ((^◡𝑈‘𝐾) − 1) ∈ (𝐸..^(((♯‘𝑈) − 1) − 1)))
167	130, 166	sseldd 3982	. . . . 5 ⊢ (𝜑 → ((^◡𝑈‘𝐾) − 1) ∈ (0..^((♯‘𝑊) − 1)))
168	1, 2, 12, 29, 167	cycpmfv1 32542	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑊‘((^◡𝑈‘𝐾) − 1))) = (𝑊‘(((^◡𝑈‘𝐾) − 1) + 1)))
169	62, 64, 63	subsub4d 11606	. . . . . . . . . 10 ⊢ (𝜑 → (((^◡𝑈‘𝐾) − 1) − 𝐸) = ((^◡𝑈‘𝐾) − (1 + 𝐸)))
170	169	oveq1d 7426	. . . . . . . . 9 ⊢ (𝜑 → ((((^◡𝑈‘𝐾) − 1) − 𝐸) + (1 + 𝐸)) = (((^◡𝑈‘𝐾) − (1 + 𝐸)) + (1 + 𝐸)))
171	64, 63	addcld 11237	. . . . . . . . . 10 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
172	62, 171	npcand 11579	. . . . . . . . 9 ⊢ (𝜑 → (((^◡𝑈‘𝐾) − (1 + 𝐸)) + (1 + 𝐸)) = (^◡𝑈‘𝐾))
173	170, 172	eqtr2d 2771	. . . . . . . 8 ⊢ (𝜑 → (^◡𝑈‘𝐾) = ((((^◡𝑈‘𝐾) − 1) − 𝐸) + (1 + 𝐸)))
174	59, 173	fveq12d 6897	. . . . . . 7 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘((((^◡𝑈‘𝐾) − 1) − 𝐸) + (1 + 𝐸))))
175	63, 75	pncan3d 11578	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 + ((♯‘𝑊) − 𝐸)) = (♯‘𝑊))
176	113, 134	eqeltrd 2831	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝑈) − 1) ∈ ℤ)
177		1zzd 12597	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℤ)
178	176, 177	zsubcld 12675	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝑈) − 1) − 1) ∈ ℤ)
179	178	zred 12670	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝑈) − 1) − 1) ∈ ℝ)
180	113, 141	eqeltrd 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((♯‘𝑈) − 1) ∈ ℝ)
181	180	ltm1d 12150	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((♯‘𝑈) − 1) − 1) < ((♯‘𝑈) − 1))
182	181, 113	breqtrd 5173	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝑈) − 1) − 1) < (♯‘𝑊))
183	179, 141, 182	ltled 11366	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝑈) − 1) − 1) ≤ (♯‘𝑊))
184		eluz1 12830	. . . . . . . . . . . . . 14 ⊢ ((((♯‘𝑈) − 1) − 1) ∈ ℤ → ((♯‘𝑊) ∈ (ℤ_≥‘(((♯‘𝑈) − 1) − 1)) ↔ ((♯‘𝑊) ∈ ℤ ∧ (((♯‘𝑈) − 1) − 1) ≤ (♯‘𝑊))))
185	184	biimpar 476	. . . . . . . . . . . . 13 ⊢ (((((♯‘𝑈) − 1) − 1) ∈ ℤ ∧ ((♯‘𝑊) ∈ ℤ ∧ (((♯‘𝑈) − 1) − 1) ≤ (♯‘𝑊))) → (♯‘𝑊) ∈ (ℤ_≥‘(((♯‘𝑈) − 1) − 1)))
186	178, 134, 183, 185	syl12anc 833	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑊) ∈ (ℤ_≥‘(((♯‘𝑈) − 1) − 1)))
187	175, 186	eqeltrd 2831	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 + ((♯‘𝑊) − 𝐸)) ∈ (ℤ_≥‘(((♯‘𝑈) − 1) − 1)))
188		fzoss2 13664	. . . . . . . . . . 11 ⊢ ((𝐸 + ((♯‘𝑊) − 𝐸)) ∈ (ℤ_≥‘(((♯‘𝑈) − 1) − 1)) → (𝐸..^(((♯‘𝑈) − 1) − 1)) ⊆ (𝐸..^(𝐸 + ((♯‘𝑊) − 𝐸))))
189	187, 188	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸..^(((♯‘𝑈) − 1) − 1)) ⊆ (𝐸..^(𝐸 + ((♯‘𝑊) − 𝐸))))
190	189, 166	sseldd 3982	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝑈‘𝐾) − 1) ∈ (𝐸..^(𝐸 + ((♯‘𝑊) − 𝐸))))
191	134, 102	zsubcld 12675	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑊) − 𝐸) ∈ ℤ)
192		fzosubel3 13697	. . . . . . . . 9 ⊢ ((((^◡𝑈‘𝐾) − 1) ∈ (𝐸..^(𝐸 + ((♯‘𝑊) − 𝐸))) ∧ ((♯‘𝑊) − 𝐸) ∈ ℤ) → (((^◡𝑈‘𝐾) − 1) − 𝐸) ∈ (0..^((♯‘𝑊) − 𝐸)))
193	190, 191, 192	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → (((^◡𝑈‘𝐾) − 1) − 𝐸) ∈ (0..^((♯‘𝑊) − 𝐸)))
194	12, 72, 39, 15, 193, 124	splfv3 32389	. . . . . . 7 ⊢ (𝜑 → ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘((((^◡𝑈‘𝐾) − 1) − 𝐸) + (1 + 𝐸))) = (𝑊‘((((^◡𝑈‘𝐾) − 1) − 𝐸) + 𝐸)))
195	62, 64	subcld 11575	. . . . . . . . 9 ⊢ (𝜑 → ((^◡𝑈‘𝐾) − 1) ∈ ℂ)
196	195, 63	npcand 11579	. . . . . . . 8 ⊢ (𝜑 → ((((^◡𝑈‘𝐾) − 1) − 𝐸) + 𝐸) = ((^◡𝑈‘𝐾) − 1))
197	196	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → (𝑊‘((((^◡𝑈‘𝐾) − 1) − 𝐸) + 𝐸)) = (𝑊‘((^◡𝑈‘𝐾) − 1)))
198	174, 194, 197	3eqtrd 2774	. . . . . 6 ⊢ (𝜑 → (𝑈‘(^◡𝑈‘𝐾)) = (𝑊‘((^◡𝑈‘𝐾) − 1)))
199	198, 57	eqtr3d 2772	. . . . 5 ⊢ (𝜑 → (𝑊‘((^◡𝑈‘𝐾) − 1)) = 𝐾)
200	199	fveq2d 6894	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑊)‘(𝑊‘((^◡𝑈‘𝐾) − 1))) = ((𝑀‘𝑊)‘𝐾))
201	62, 64	npcand 11579	. . . . 5 ⊢ (𝜑 → (((^◡𝑈‘𝐾) − 1) + 1) = (^◡𝑈‘𝐾))
202	201	fveq2d 6894	. . . 4 ⊢ (𝜑 → (𝑊‘(((^◡𝑈‘𝐾) − 1) + 1)) = (𝑊‘(^◡𝑈‘𝐾)))
203	168, 200, 202	3eqtr3d 2778	. . 3 ⊢ (𝜑 → ((𝑀‘𝑊)‘𝐾) = (𝑊‘(^◡𝑈‘𝐾)))
204	70, 125, 203	3eqtr4rd 2781	. 2 ⊢ (𝜑 → ((𝑀‘𝑊)‘𝐾) = ((𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)‘(((^◡𝑈‘𝐾) − 𝐸) + (1 + 𝐸))))
205	68, 204	eqtr4d 2773	1 ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 {crab 3430 Vcvv 3472 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 {csn 4627 ⟨cop 4633 ⟨cotp 4635 class class class wbr 5147 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ⟶wf 6538 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13488 ..^cfzo 13631 ♯chash 14294 Word cword 14468 ++ cconcat 14524 ⟨“cs1 14549 substr csubstr 14594 prefix cpfx 14624 splice csplice 14703 Basecbs 17148 SymGrpcsymg 19275 toCycctocyc 32535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14704 df-csh 14743 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-tset 17220 df-efmnd 18786 df-symg 19276 df-tocyc 32536

This theorem is referenced by: cycpmco2 32562

W3C validator