cycpmrn - Mathbox for Thierry Arnoux

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Theorem cycpmrn 33219

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 733	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑊:dom 𝑊–1-1→𝐷)
3		simpllr 776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ dom 𝑊)
4		fzo0ss1 13635	. . . . . . . . . . . . 13 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))
5		simpr 484	. . . . . . . . . . . . . 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 14486	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ₀)
9	8	ad4antr 733	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (♯‘𝑊) ∈ ℕ₀)
10	9	nn0zd 12540	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (♯‘𝑊) ∈ ℤ)
11		1zzd 12549	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 1 ∈ ℤ)
12		fzoaddel2 13666	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∧ (♯‘𝑊) ∈ ℤ ∧ 1 ∈ ℤ) → (𝑥 + 1) ∈ (1..^(♯‘𝑊)))
13	5, 10, 11, 12	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ (1..^(♯‘𝑊)))
14	4, 13	sselid 3920	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ (0..^(♯‘𝑊)))
15	6	ad4antr 733	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑊 ∈ Word 𝐷)
16		wrddm 14474	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → dom 𝑊 = (0..^(♯‘𝑊)))
18	14, 17	eleqtrrd 2840	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ dom 𝑊)
19		fzossz 13625	. . . . . . . . . . . . . 14 ⊢ (0..^((♯‘𝑊) − 1)) ⊆ ℤ
20	19, 5	sselid 3920	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ ℤ)
21	20	zred 12624	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ ℝ)
22	21	ltp1d 12077	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 < (𝑥 + 1))
23	21, 22	ltned 11273	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ≠ (𝑥 + 1))
24		f1veqaeq 7204	. . . . . . . . . . . . . 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 2954	. . . . . . . . . . . . 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 467	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 841	. . . . . . . . . 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 733	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝐷 ∈ 𝑉)
32	29, 31, 15, 2, 5	cycpmfv1 33189	. . . . . . . . . 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 3007	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑊‘𝑥) ≠ ((𝑀‘𝑊)‘(𝑊‘𝑥)))
34	33	necomd 2988	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑀‘𝑊)‘(𝑊‘𝑥)) ≠ (𝑊‘𝑥))
35		simplr 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑦 = (𝑊‘𝑥))
36	35	fveq2d 6838	. . . . . . . 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 3010	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
38	1	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑊:dom 𝑊–1-1→𝐷)
39	6	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑊 ∈ Word 𝐷)
40		eldmne0 32715	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑊 → 𝑊 ≠ ∅)
41	40	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑊 ≠ ∅)
42		lennncl 14487	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)
43	39, 41, 42	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (♯‘𝑊) ∈ ℕ)
44		lbfzo0 13645	. . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
45	43, 44	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 0 ∈ (0..^(♯‘𝑊)))
46	39, 16	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → dom 𝑊 = (0..^(♯‘𝑊)))
47	45, 46	eleqtrrd 2840	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 0 ∈ dom 𝑊)
48	47	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 ∈ dom 𝑊)
49		simpllr 776	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑥 ∈ dom 𝑊)
50		0red 11138	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
51		cycpmrn.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < (♯‘𝑊))
52		1red 11136	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
53	8	nn0red 12490	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
54	52, 53	posdifd 11728	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < (♯‘𝑊) ↔ 0 < ((♯‘𝑊) − 1)))
55	51, 54	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((♯‘𝑊) − 1))
56	50, 55	ltned 11273	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≠ ((♯‘𝑊) − 1))
57	56	ad4antr 733	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 ≠ ((♯‘𝑊) − 1))
58		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑥 = ((♯‘𝑊) − 1))
59	57, 58	neeqtrrd 3007	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 ≠ 𝑥)
60		f1veqaeq 7204	. . . . . . . . . . . 12 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ (0 ∈ dom 𝑊 ∧ 𝑥 ∈ dom 𝑊)) → ((𝑊‘0) = (𝑊‘𝑥) → 0 = 𝑥))
61	60	necon3d 2954	. . . . . . . . . . 11 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ (0 ∈ dom 𝑊 ∧ 𝑥 ∈ dom 𝑊)) → (0 ≠ 𝑥 → (𝑊‘0) ≠ (𝑊‘𝑥)))
62	61	anassrs 467	. . . . . . . . . 10 ⊢ (((𝑊:dom 𝑊–1-1→𝐷 ∧ 0 ∈ dom 𝑊) ∧ 𝑥 ∈ dom 𝑊) → (0 ≠ 𝑥 → (𝑊‘0) ≠ (𝑊‘𝑥)))
63	62	imp 406	. . . . . . . . 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 841	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → (𝑊‘0) ≠ (𝑊‘𝑥))
65		simplr 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑦 = (𝑊‘𝑥))
66	65	fveq2d 6838	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) = ((𝑀‘𝑊)‘(𝑊‘𝑥)))
67	30	ad4antr 733	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝐷 ∈ 𝑉)
68	6	ad4antr 733	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 𝑊 ∈ Word 𝐷)
69	43	nngt0d 12217	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 0 < (♯‘𝑊))
70	69	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → 0 < (♯‘𝑊))
71	29, 67, 68, 38, 70, 58	cycpmfv2 33190	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘(𝑊‘𝑥)) = (𝑊‘0))
72	66, 71	eqtrd 2772	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) = (𝑊‘0))
73	64, 72, 65	3netr4d 3010	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
74		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ dom 𝑊)
75	74, 46	eleqtrd 2839	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ (0..^(♯‘𝑊)))
76		0z 12526	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
77		0p1e1 12289	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
78	77	fveq2i 6837	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
79		nnuz 12818	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eqtr4i 2763	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
81	43, 80	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (♯‘𝑊) ∈ (ℤ_≥‘(0 + 1)))
82		fzosplitsnm1 13686	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ (♯‘𝑊) ∈ (ℤ_≥‘(0 + 1))) → (0..^(♯‘𝑊)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
83	76, 81, 82	sylancr 588	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (0..^(♯‘𝑊)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
84	75, 83	eleqtrd 2839	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
85		elun 4094	. . . . . . . . 9 ⊢ (𝑥 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}) ↔ (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 ∈ {((♯‘𝑊) − 1)}))
86	84, 85	sylib 218	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 ∈ {((♯‘𝑊) − 1)}))
87		velsn 4584	. . . . . . . . 9 ⊢ (𝑥 ∈ {((♯‘𝑊) − 1)} ↔ 𝑥 = ((♯‘𝑊) − 1))
88	87	orbi2i 913	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 ∈ {((♯‘𝑊) − 1)}) ↔ (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 = ((♯‘𝑊) − 1)))
89	86, 88	sylib 218	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (𝑥 ∈ (0..^((♯‘𝑊) − 1)) ∨ 𝑥 = ((♯‘𝑊) − 1)))
90	37, 73, 89	mpjaodan 961	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
91		f1fun 6732	. . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → Fun 𝑊)
92		elrnrexdmb 7036	. . . . . . . 8 ⊢ (Fun 𝑊 → (𝑦 ∈ ran 𝑊 ↔ ∃𝑥 ∈ dom 𝑊 𝑦 = (𝑊‘𝑥)))
93	1, 91, 92	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ran 𝑊 ↔ ∃𝑥 ∈ dom 𝑊 𝑦 = (𝑊‘𝑥)))
94	93	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → ∃𝑥 ∈ dom 𝑊 𝑦 = (𝑊‘𝑥))
95	90, 94	r19.29a 3146	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
96		eqid 2737	. . . . . . . . . 10 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
97	29, 30, 6, 1, 96	cycpmcl 33192	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)))
98		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
99	96, 98	elsymgbas 19340	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → ((𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)) ↔ (𝑀‘𝑊):𝐷–1-1-onto→𝐷))
100	30, 99	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)) ↔ (𝑀‘𝑊):𝐷–1-1-onto→𝐷))
101	97, 100	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝑊):𝐷–1-1-onto→𝐷)
102		f1ofn 6775	. . . . . . . 8 ⊢ ((𝑀‘𝑊):𝐷–1-1-onto→𝐷 → (𝑀‘𝑊) Fn 𝐷)
103	101, 102	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝑊) Fn 𝐷)
104	103	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → (𝑀‘𝑊) Fn 𝐷)
105		wrdf 14471	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊:(0..^(♯‘𝑊))⟶𝐷)
106		frn 6669	. . . . . . . 8 ⊢ (𝑊:(0..^(♯‘𝑊))⟶𝐷 → ran 𝑊 ⊆ 𝐷)
107	6, 105, 106	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
108	107	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → 𝑦 ∈ 𝐷)
109		fnelnfp 7125	. . . . . 6 ⊢ (((𝑀‘𝑊) Fn 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑦 ∈ dom ((𝑀‘𝑊) ∖ I ) ↔ ((𝑀‘𝑊)‘𝑦) ≠ 𝑦))
110	104, 108, 109	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → (𝑦 ∈ dom ((𝑀‘𝑊) ∖ I ) ↔ ((𝑀‘𝑊)‘𝑦) ≠ 𝑦))
111	95, 110	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → 𝑦 ∈ dom ((𝑀‘𝑊) ∖ I ))
112	111	ex 412	. . 3 ⊢ (𝜑 → (𝑦 ∈ ran 𝑊 → 𝑦 ∈ dom ((𝑀‘𝑊) ∖ I )))
113	112	ssrdv 3928	. 2 ⊢ (𝜑 → ran 𝑊 ⊆ dom ((𝑀‘𝑊) ∖ I ))
114	29, 30, 6, 1	tocycfv 33185	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
115	114	difeq1d 4066	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑊) ∖ I ) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ))
116	115	dmeqd 5854	. . 3 ⊢ (𝜑 → dom ((𝑀‘𝑊) ∖ I ) = dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ))
117		difundir 4232	. . . . . 6 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ))
118		resdifcom 5957	. . . . . . . 8 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) = (( I ∖ I ) ↾ (𝐷 ∖ ran 𝑊))
119		difid 4317	. . . . . . . . 9 ⊢ ( I ∖ I ) = ∅
120	119	reseq1i 5934	. . . . . . . 8 ⊢ (( I ∖ I ) ↾ (𝐷 ∖ ran 𝑊)) = (∅ ↾ (𝐷 ∖ ran 𝑊))
121		0res 32688	. . . . . . . 8 ⊢ (∅ ↾ (𝐷 ∖ ran 𝑊)) = ∅
122	118, 120, 121	3eqtri 2764	. . . . . . 7 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) = ∅
123	122	uneq1i 4105	. . . . . 6 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )) = (∅ ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ))
124		0un 4337	. . . . . 6 ⊢ (∅ ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )) = (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
125	117, 123, 124	3eqtri 2764	. . . . 5 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
126	125	dmeqi 5853	. . . 4 ⊢ dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
127		difss 4077	. . . . . 6 ⊢ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ((𝑊 cyclShift 1) ∘ ^◡𝑊)
128		dmss 5851	. . . . . 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 5924	. . . . . 6 ⊢ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊) ⊆ dom ^◡𝑊
131		df-rn 5635	. . . . . 6 ⊢ ran 𝑊 = dom ^◡𝑊
132	130, 131	sseqtrri 3972	. . . . 5 ⊢ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊) ⊆ ran 𝑊
133	129, 132	sstri 3932	. . . 4 ⊢ dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ran 𝑊
134	126, 133	eqsstri 3969	. . 3 ⊢ dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) ⊆ ran 𝑊
135	116, 134	eqsstrdi 3967	. 2 ⊢ (𝜑 → dom ((𝑀‘𝑊) ∖ I ) ⊆ ran 𝑊)
136	113, 135	eqssd 3940	1 ⊢ (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ∖ cdif 3887 ∪ cun 3888 ⊆ wss 3890 ∅c0 4274 {csn 4568 class class class wbr 5086 I cid 5518 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 ∘ ccom 5628 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 0cc0 11029 1c1 11030 + caddc 11032 < clt 11170 − cmin 11368 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ..^cfzo 13599 ♯chash 14283 Word cword 14466 cyclShift ccsh 14741 Basecbs 17170 SymGrpcsymg 19335 toCycctocyc 33182

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-inf 9349 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-hash 14284 df-word 14467 df-concat 14524 df-substr 14595 df-pfx 14625 df-csh 14742 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-tset 17230 df-efmnd 18828 df-symg 19336 df-tocyc 33183

This theorem is referenced by: tocyccntz 33220

W3C validator