cycpmrn - Mathbox for Thierry Arnoux

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Theorem cycpmrn 33705

Description: The range of the word used to build a cycle is the cycle's orbit, i.e., the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.)

**Hypotheses**
Ref	Expression
cycpmrn.1	⊢ 𝑀 = (toCyc‘𝐷)
cycpmrn.2	⊢ (𝜑 → 𝐷 ∈ 𝑉)
cycpmrn.3	⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
cycpmrn.4	⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
cycpmrn.5	⊢ (𝜑 → 1 < (♯‘𝑊))

**Assertion**
Ref	Expression
cycpmrn	⊢ (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I ))

Proof of Theorem cycpmrn Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmrn.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
2	1	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑊:dom 𝑊–1-1→𝐷)
3		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ dom 𝑊)
4		fzo0ss1 13825	. . . . . . . . . . . . 13 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))
5		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ (0..^((♯‘𝑊) − 1)))
6		cycpmrn.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
7		lencl 14679	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
8	6, 7	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
9	8	ad4antr 745	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (♯‘𝑊) ∈ ℕ₀)
10	9	nn0zd 12719	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (♯‘𝑊) ∈ ℤ)
11		1zzd 12728	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 1 ∈ ℤ)
12		fzoaddel2 13856	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∧ (♯‘𝑊) ∈ ℤ ∧ 1 ∈ ℤ) → (𝑥 + 1) ∈ (1..^(♯‘𝑊)))
13	5, 10, 11, 12	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ (1..^(♯‘𝑊)))
14	4, 13	sselid 3927	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ (0..^(♯‘𝑊)))
15	6	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑊 ∈ Word 𝐷)
16		wrddm 14667	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
17	15, 16	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → dom 𝑊 = (0..^(♯‘𝑊)))
18	14, 17	eleqtrrd 2864	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ dom 𝑊)
19		fzossz 13815	. . . . . . . . . . . . . 14 ⊢ (0..^((♯‘𝑊) − 1)) ⊆ ℤ
20	19, 5	sselid 3927	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ ℤ)
21	20	zred 12804	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ ℝ)
22	21	ltp1d 12248	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 < (𝑥 + 1))
23	21, 22	ltned 11447	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ≠ (𝑥 + 1))
24		f1veqaeq 7261	. . . . . . . . . . . . . 14 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ (𝑥 ∈ dom 𝑊 ∧ (𝑥 + 1) ∈ dom 𝑊)) → ((𝑊‘𝑥) = (𝑊‘(𝑥 + 1)) → 𝑥 = (𝑥 + 1)))
25	24	necon3d 2977	. . . . . . . . . . . . 13 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ (𝑥 ∈ dom 𝑊 ∧ (𝑥 + 1) ∈ dom 𝑊)) → (𝑥 ≠ (𝑥 + 1) → (𝑊‘𝑥) ≠ (𝑊‘(𝑥 + 1))))
26	25	anassrs 473	. . . . . . . . . . . 12 ⊢ (((𝑊:dom 𝑊–1-1→𝐷 ∧ 𝑥 ∈ dom 𝑊) ∧ (𝑥 + 1) ∈ dom 𝑊) → (𝑥 ≠ (𝑥 + 1) → (𝑊‘𝑥) ≠ (𝑊‘(𝑥 + 1))))
27	26	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑊:dom 𝑊–1-1→𝐷 ∧ 𝑥 ∈ dom 𝑊) ∧ (𝑥 + 1) ∈ dom 𝑊) ∧ 𝑥 ≠ (𝑥 + 1)) → (𝑊‘𝑥) ≠ (𝑊‘(𝑥 + 1)))
28	2, 3, 18, 23, 27	syl1111anc 854	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑊‘𝑥) ≠ (𝑊‘(𝑥 + 1)))
29		cycpmrn.1	. . . . . . . . . . 11 ⊢ 𝑀 = (toCyc‘𝐷)
30		cycpmrn.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
31	30	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝐷 ∈ 𝑉)
32	29, 31, 15, 2, 5	cycpmfv1 33675	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑀‘𝑊)‘(𝑊‘𝑥)) = (𝑊‘(𝑥 + 1)))
33	28, 32	neeqtrrd 3030	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑊‘𝑥) ≠ ((𝑀‘𝑊)‘(𝑊‘𝑥)))
34	33	necomd 3011	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑀‘𝑊)‘(𝑊‘𝑥)) ≠ (𝑊‘𝑥))
35		simplr 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑦 = (𝑊‘𝑥))
36	35	fveq2d 6890	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑀‘𝑊)‘𝑦) = ((𝑀‘𝑊)‘(𝑊‘𝑥)))
37	34, 36, 35	3netr4d 3033	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
38	1	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑊:dom 𝑊–1-1→𝐷)
39	6	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑊 ∈ Word 𝐷)
40		eldmne0 33222	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑊 → 𝑊 ≠ ∅)
41	40	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑊 ≠ ∅)
42		lennncl 14680	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)
43	39, 41, 42	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (♯‘𝑊) ∈ ℕ)
44		lbfzo0 13835	. . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
45	43, 44	sylibr 237	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 0 ∈ (0..^(♯‘𝑊)))
46	39, 16	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → dom 𝑊 = (0..^(♯‘𝑊)))
47	45, 46	eleqtrrd 2864	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 0 ∈ dom 𝑊)
48	47	adantr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 ∈ dom 𝑊)
49		simpllr 788	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑥 ∈ dom 𝑊)
50		0red 11312	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
51		cycpmrn.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < (♯‘𝑊))
52		1red 11310	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
53	8	nn0red 12669	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
54	52, 53	posdifd 11904	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < (♯‘𝑊) ↔ 0 < ((♯‘𝑊) − 1)))
55	51, 54	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((♯‘𝑊) − 1))
56	50, 55	ltned 11447	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≠ ((♯‘𝑊) − 1))
57	56	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 ≠ ((♯‘𝑊) − 1))
58		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑥 = ((♯‘𝑊) − 1))
59	57, 58	neeqtrrd 3030	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 ≠ 𝑥)
60		f1veqaeq 7261	. . . . . . . . . . . 12 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ (0 ∈ dom 𝑊 ∧ 𝑥 ∈ dom 𝑊)) → ((𝑊‘0) = (𝑊‘𝑥) → 0 = 𝑥))
61	60	necon3d 2977	. . . . . . . . . . 11 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ (0 ∈ dom 𝑊 ∧ 𝑥 ∈ dom 𝑊)) → (0 ≠ 𝑥 → (𝑊‘0) ≠ (𝑊‘𝑥)))
62	61	anassrs 473	. . . . . . . . . 10 ⊢ (((𝑊:dom 𝑊–1-1→𝐷 ∧ 0 ∈ dom 𝑊) ∧ 𝑥 ∈ dom 𝑊) → (0 ≠ 𝑥 → (𝑊‘0) ≠ (𝑊‘𝑥)))
63	62	imp 412	. . . . . . . . 9 ⊢ ((((𝑊:dom 𝑊–1-1→𝐷 ∧ 0 ∈ dom 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 0 ≠ 𝑥) → (𝑊‘0) ≠ (𝑊‘𝑥))
64	38, 48, 49, 59, 63	syl1111anc 854	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → (𝑊‘0) ≠ (𝑊‘𝑥))
65		simplr 781	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑦 = (𝑊‘𝑥))
66	65	fveq2d 6890	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) = ((𝑀‘𝑊)‘(𝑊‘𝑥)))
67	30	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝐷 ∈ 𝑉)
68	6	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑊 ∈ Word 𝐷)
69	43	nngt0d 12388	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 0 < (♯‘𝑊))
70	69	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 < (♯‘𝑊))
71	29, 67, 68, 38, 70, 58	cycpmfv2 33676	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘(𝑊‘𝑥)) = (𝑊‘0))
72	66, 71	eqtrd 2796	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) = (𝑊‘0))
73	64, 72, 65	3netr4d 3033	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
74		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ dom 𝑊)
75	74, 46	eleqtrd 2863	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ (0..^(♯‘𝑊)))
76		0z 12705	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
77		0p1e1 12464	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
78	77	fveq2i 6889	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
79		nnuz 13005	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eqtr4i 2787	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
81	43, 80	eleqtrrdi 2872	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (♯‘𝑊) ∈ (ℤ_≥‘(0 + 1)))
82		fzosplitsnm1 13876	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ (♯‘𝑊) ∈ (ℤ_≥‘(0 + 1))) → (0..^(♯‘𝑊)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
83	76, 81, 82	sylancr 599	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (0..^(♯‘𝑊)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
84	75, 83	eleqtrd 2863	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
85		elun 4100	. . . . . . . . 9 ⊢ (𝑥 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}) ↔ (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 ∈ {((♯‘𝑊) − 1)}))
86	84, 85	sylib 221	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 ∈ {((♯‘𝑊) − 1)}))
87		velsn 4600	. . . . . . . . 9 ⊢ (𝑥 ∈ {((♯‘𝑊) − 1)} ↔ 𝑥 = ((♯‘𝑊) − 1))
88	87	orbi2i 926	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 ∈ {((♯‘𝑊) − 1)}) ↔ (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 = ((♯‘𝑊) − 1)))
89	86, 88	sylib 221	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 = ((♯‘𝑊) − 1)))
90	37, 73, 89	mpjaodan 973	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
91		f1fun 6781	. . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → Fun 𝑊)
92		elrnrexdmb 7091	. . . . . . . 8 ⊢ (Fun 𝑊 → (𝑦 ∈ ran 𝑊 ↔ ∃𝑥 ∈ dom 𝑊 𝑦 = (𝑊‘𝑥)))
93	1, 91, 92	3syl 19	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran 𝑊 ↔ ∃𝑥 ∈ dom 𝑊 𝑦 = (𝑊‘𝑥)))
94	93	biimpa 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → ∃𝑥 ∈ dom 𝑊 𝑦 = (𝑊‘𝑥))
95	90, 94	r19.29a 3171	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
96		eqid 2761	. . . . . . . . . 10 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
97	29, 30, 6, 1, 96	cycpmcl 33678	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)))
98		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
99	96, 98	elsymgbas 19589	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → ((𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)) ↔ (𝑀‘𝑊):𝐷–1-1-onto→𝐷))
100	30, 99	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)) ↔ (𝑀‘𝑊):𝐷–1-1-onto→𝐷))
101	97, 100	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝑊):𝐷–1-1-onto→𝐷)
102		f1ofn 6826	. . . . . . . 8 ⊢ ((𝑀‘𝑊):𝐷–1-1-onto→𝐷 → (𝑀‘𝑊) Fn 𝐷)
103	101, 102	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝑊) Fn 𝐷)
104	103	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → (𝑀‘𝑊) Fn 𝐷)
105		wrdf 14664	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊:(0..^(♯‘𝑊))⟶𝐷)
106		frn 6718	. . . . . . . 8 ⊢ (𝑊:(0..^(♯‘𝑊))⟶𝐷 → ran 𝑊 ⊆ 𝐷)
107	6, 105, 106	3syl 19	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
108	107	sselda 3929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → 𝑦 ∈ 𝐷)
109		fnelnfp 7183	. . . . . 6 ⊢ (((𝑀‘𝑊) Fn 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑦 ∈ dom ((𝑀‘𝑊) ∖ I ) ↔ ((𝑀‘𝑊)‘𝑦) ≠ 𝑦))
110	104, 108, 109	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → (𝑦 ∈ dom ((𝑀‘𝑊) ∖ I ) ↔ ((𝑀‘𝑊)‘𝑦) ≠ 𝑦))
111	95, 110	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → 𝑦 ∈ dom ((𝑀‘𝑊) ∖ I ))
112	111	ex 418	. . 3 ⊢ (𝜑 → (𝑦 ∈ ran 𝑊 → 𝑦 ∈ dom ((𝑀‘𝑊) ∖ I )))
113	112	ssrdv 3935	. 2 ⊢ (𝜑 → ran 𝑊 ⊆ dom ((𝑀‘𝑊) ∖ I ))
114	29, 30, 6, 1	tocycfv 33671	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
115	114	difeq1d 4073	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑊) ∖ I ) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ))
116	115	dmeqd 5887	. . 3 ⊢ (𝜑 → dom ((𝑀‘𝑊) ∖ I ) = dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ))
117		difundir 4237	. . . . . 6 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ))
118		resdifcom 5989	. . . . . . . 8 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) = (( I ∖ I ) ↾ (𝐷 ∖ ran 𝑊))
119		difid 4325	. . . . . . . . 9 ⊢ ( I ∖ I ) = ∅
120	119	reseq1i 5966	. . . . . . . 8 ⊢ (( I ∖ I ) ↾ (𝐷 ∖ ran 𝑊)) = (∅ ↾ (𝐷 ∖ ran 𝑊))
121		0res 33198	. . . . . . . 8 ⊢ (∅ ↾ (𝐷 ∖ ran 𝑊)) = ∅
122	118, 120, 121	3eqtri 2788	. . . . . . 7 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) = ∅
123	122	uneq1i 4111	. . . . . 6 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )) = (∅ ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ))
124		0un 4346	. . . . . 6 ⊢ (∅ ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )) = (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
125	117, 123, 124	3eqtri 2788	. . . . 5 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
126	125	dmeqi 5886	. . . 4 ⊢ dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
127		difss 4083	. . . . . 6 ⊢ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ((𝑊 cyclShift 1) ∘ ^◡𝑊)
128		dmss 5884	. . . . . 6 ⊢ ((((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ((𝑊 cyclShift 1) ∘ ^◡𝑊) → dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊))
129	127, 128	ax-mp 5	. . . . 5 ⊢ dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊)
130		dmcoss 5957	. . . . . 6 ⊢ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊) ⊆ dom ^◡𝑊
131		df-rn 5662	. . . . . 6 ⊢ ran 𝑊 = dom ^◡𝑊
132	130, 131	sseqtrri 3978	. . . . 5 ⊢ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊) ⊆ ran 𝑊
133	129, 132	sstri 3938	. . . 4 ⊢ dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ran 𝑊
134	126, 133	eqsstri 3975	. . 3 ⊢ dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) ⊆ ran 𝑊
135	116, 134	eqsstrdi 3973	. 2 ⊢ (𝜑 → dom ((𝑀‘𝑊) ∖ I ) ⊆ ran 𝑊)
136	113, 135	eqssd 3946	1 ⊢ (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3914 ∅c0 4279 {csn 4584 class class class wbr 5103 I cid 5545 ^◡ccnv 5650 dom cdm 5651 ran crn 5652 ↾ cres 5653 ∘ ccom 5655 Fun wfun 6533 Fn wfn 6534 ⟶wf 6535 –1-1→wf1 6536 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7421 0cc0 11201 1c1 11202 + caddc 11204 < clt 11344 − cmin 11542 ℕcn 12336 ℕ₀cn0 12607 ℤcz 12694 ℤ_≥cuz 12966 ..^cfzo 13789 ♯chash 14475 Word cword 14659 cyclShift ccsh 14940 Basecbs 17388 SymGrpcsymg 19584 toCycctocyc 33668

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9435 df-inf 9436 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-z 12695 df-uz 12967 df-rp 13122 df-fz 13641 df-fzo 13790 df-fl 13933 df-mod 14011 df-hash 14476 df-word 14660 df-concat 14717 df-substr 14790 df-pfx 14822 df-csh 14941 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-tset 17448 df-efmnd 19066 df-symg 19585 df-tocyc 33669

This theorem is used by: tocyccntz 33706

W3C validator