cycpmrn - Mathbox for Thierry Arnoux

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Theorem cycpmrn 33494

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 13737	. . . . . . . . . . . . 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 14590	. . . . . . . . . . . . . . . . 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 12634	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (♯‘𝑊) ∈ ℤ)
11		1zzd 12643	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 1 ∈ ℤ)
12		fzoaddel2 13768	. . . . . . . . . . . . . 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 3938	. . . . . . . . . . . 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 14578	. . . . . . . . . . . . 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 2869	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑥 + 1) ∈ dom 𝑊)
19		fzossz 13727	. . . . . . . . . . . . . 14 ⊢ (0..^((♯‘𝑊) − 1)) ⊆ ℤ
20	19, 5	sselid 3938	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ ℤ)
21	20	zred 12718	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 ∈ ℝ)
22	21	ltp1d 12163	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → 𝑥 < (𝑥 + 1))
23	21, 22	ltned 11364	. . . . . . . . . . 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 2982	. . . . . . . . . . . . 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 33464	. . . . . . . . . 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 3035	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 ∈ (0..^((♯‘𝑊) − 1))) → (𝑊‘𝑥) ≠ ((𝑀‘𝑊)‘(𝑊‘𝑥)))
34	33	necomd 3016	. . . . . . . 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 6892	. . . . . . . 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 3038	. . . . . . 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 33009	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑊 → 𝑊 ≠ ∅)
41	40	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑊 ≠ ∅)
42		lennncl 14591	. . . . . . . . . . . . 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 13747	. . . . . . . . . . . 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 2869	. . . . . . . . . 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 11229	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
51		cycpmrn.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < (♯‘𝑊))
52		1red 11227	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
53	8	nn0red 12584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑊) ∈ ℝ)
54	52, 53	posdifd 11819	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < (♯‘𝑊) ↔ 0 < ((♯‘𝑊) − 1)))
55	51, 54	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((♯‘𝑊) − 1))
56	50, 55	ltned 11364	. . . . . . . . . . 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 3035	. . . . . . . . 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 2982	. . . . . . . . . . 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 6892	. . . . . . . . 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 12303	. . . . . . . . . . 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 33465	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘(𝑊‘𝑥)) = (𝑊‘0))
72	66, 71	eqtrd 2801	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) ∧ 𝑥 = ((♯‘𝑊) − 1)) → ((𝑀‘𝑊)‘𝑦) = (𝑊‘0))
73	64, 72, 65	3netr4d 3038	. . . . . . 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 2868	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ (0..^(♯‘𝑊)))
76		0z 12620	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
77		0p1e1 12379	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
78	77	fveq2i 6891	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
79		nnuz 12919	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eqtr4i 2792	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
81	43, 80	eleqtrrdi 2877	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → (♯‘𝑊) ∈ (ℤ_≥‘(0 + 1)))
82		fzosplitsnm1 13788	. . . . . . . . . . 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 2868	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝑊) ∧ 𝑥 ∈ dom 𝑊) ∧ 𝑦 = (𝑊‘𝑥)) → 𝑥 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1)}))
85		elun 4110	. . . . . . . . 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 4610	. . . . . . . . 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 6783	. . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → Fun 𝑊)
92		elrnrexdmb 7092	. . . . . . . 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 3176	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → ((𝑀‘𝑊)‘𝑦) ≠ 𝑦)
96		eqid 2766	. . . . . . . . . 10 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷)
97	29, 30, 6, 1, 96	cycpmcl 33467	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘𝑊) ∈ (Base‘(SymGrp‘𝐷)))
98		eqid 2766	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘𝐷)) = (Base‘(SymGrp‘𝐷))
99	96, 98	elsymgbas 19475	. . . . . . . . . 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 6828	. . . . . . . 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 14575	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊:(0..^(♯‘𝑊))⟶𝐷)
106		frn 6720	. . . . . . . 8 ⊢ (𝑊:(0..^(♯‘𝑊))⟶𝐷 → ran 𝑊 ⊆ 𝐷)
107	6, 105, 106	3syl 19	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
108	107	sselda 3940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑊) → 𝑦 ∈ 𝐷)
109		fnelnfp 7182	. . . . . 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 3946	. 2 ⊢ (𝜑 → ran 𝑊 ⊆ dom ((𝑀‘𝑊) ∖ I ))
114	29, 30, 6, 1	tocycfv 33460	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)))
115	114	difeq1d 4083	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑊) ∖ I ) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ))
116	115	dmeqd 5900	. . 3 ⊢ (𝜑 → dom ((𝑀‘𝑊) ∖ I ) = dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ))
117		difundir 4247	. . . . . 6 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ))
118		resdifcom 6002	. . . . . . . 8 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) = (( I ∖ I ) ↾ (𝐷 ∖ ran 𝑊))
119		difid 4335	. . . . . . . . 9 ⊢ ( I ∖ I ) = ∅
120	119	reseq1i 5979	. . . . . . . 8 ⊢ (( I ∖ I ) ↾ (𝐷 ∖ ran 𝑊)) = (∅ ↾ (𝐷 ∖ ran 𝑊))
121		0res 32985	. . . . . . . 8 ⊢ (∅ ↾ (𝐷 ∖ ran 𝑊)) = ∅
122	118, 120, 121	3eqtri 2793	. . . . . . 7 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) = ∅
123	122	uneq1i 4121	. . . . . 6 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∖ I ) ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )) = (∅ ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ))
124		0un 4356	. . . . . 6 ⊢ (∅ ∪ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )) = (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
125	117, 123, 124	3eqtri 2793	. . . . 5 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
126	125	dmeqi 5899	. . . 4 ⊢ dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) = dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I )
127		difss 4093	. . . . . 6 ⊢ (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ((𝑊 cyclShift 1) ∘ ^◡𝑊)
128		dmss 5897	. . . . . 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 5970	. . . . . 6 ⊢ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊) ⊆ dom ^◡𝑊
131		df-rn 5677	. . . . . 6 ⊢ ran 𝑊 = dom ^◡𝑊
132	130, 131	sseqtrri 3989	. . . . 5 ⊢ dom ((𝑊 cyclShift 1) ∘ ^◡𝑊) ⊆ ran 𝑊
133	129, 132	sstri 3949	. . . 4 ⊢ dom (((𝑊 cyclShift 1) ∘ ^◡𝑊) ∖ I ) ⊆ ran 𝑊
134	126, 133	eqsstri 3986	. . 3 ⊢ dom ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ^◡𝑊)) ∖ I ) ⊆ ran 𝑊
135	116, 134	eqsstrdi 3984	. 2 ⊢ (𝜑 → dom ((𝑀‘𝑊) ∖ I ) ⊆ ran 𝑊)
136	113, 135	eqssd 3957	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 2146 ≠ wne 2961 ∃wrex 3092 ∖ cdif 3905 ∪ cun 3906 ⊆ wss 3908 ∅c0 4289 {csn 4594 class class class wbr 5114 I cid 5560 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 ↾ cres 5668 ∘ ccom 5670 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7423 0cc0 11118 1c1 11119 + caddc 11121 < clt 11261 − cmin 11459 ℕcn 12251 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ..^cfzo 13701 ♯chash 14386 Word cword 14570 cyclShift ccsh 14851 Basecbs 17294 SymGrpcsymg 19470 toCycctocyc 33457

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 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 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-hash 14387 df-word 14571 df-concat 14628 df-substr 14701 df-pfx 14733 df-csh 14852 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-tset 17354 df-efmnd 18959 df-symg 19471 df-tocyc 33458

This theorem is used by: tocyccntz 33495

W3C validator