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Theorem cshweqrep 14223

Description: If cyclically shifting a word by L position results in the word itself, the symbol at any position is repeated at multiples of L (modulo the length of the word) positions in the word. (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Revised by AV, 1-Nov-2018.)

**Assertion**
Ref	Expression
cshweqrep	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ∀𝑗 ∈ ℕ₀ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))

Distinct variable groups: 𝑗,𝐼 𝑗,𝐿 𝑗,𝑉 𝑗,𝑊

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

Step	Hyp	Ref	Expression
1		oveq1 7158	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 · 𝐿) = (0 · 𝐿))
2	1	oveq2d 7167	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (0 · 𝐿)))
3	2	fvoveq1d 7173	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))
4	3	eqeq2d 2770	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)))))
5	4	imbi2d 345	. . . . 5 ⊢ (𝑥 = 0 → ((((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊)))) ↔ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))))
6		oveq1 7158	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐿) = (𝑦 · 𝐿))
7	6	oveq2d 7167	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (𝑦 · 𝐿)))
8	7	fvoveq1d 7173	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
9	8	eqeq2d 2770	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)))))
10	9	imbi2d 345	. . . . 5 ⊢ (𝑥 = 𝑦 → ((((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊)))) ↔ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))))
11		oveq1 7158	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝐿) = ((𝑦 + 1) · 𝐿))
12	11	oveq2d 7167	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + ((𝑦 + 1) · 𝐿)))
13	12	fvoveq1d 7173	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
14	13	eqeq2d 2770	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
15	14	imbi2d 345	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊)))) ↔ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))))
16		oveq1 7158	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (𝑥 · 𝐿) = (𝑗 · 𝐿))
17	16	oveq2d 7167	. . . . . . . 8 ⊢ (𝑥 = 𝑗 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (𝑗 · 𝐿)))
18	17	fvoveq1d 7173	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊))))
19	18	eqeq2d 2770	. . . . . 6 ⊢ (𝑥 = 𝑗 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))
20	19	imbi2d 345	. . . . 5 ⊢ (𝑥 = 𝑗 → ((((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊)))) ↔ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊))))))
21		zcn 12018	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
22	21	mul02d 10869	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℤ → (0 · 𝐿) = 0)
23	22	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (0 · 𝐿) = 0)
24	23	adantr 485	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (0 · 𝐿) = 0)
25	24	oveq2d 7167	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + (0 · 𝐿)) = (𝐼 + 0))
26		elfzoelz 13080	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → 𝐼 ∈ ℤ)
27	26	zcnd 12120	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → 𝐼 ∈ ℂ)
28	27	addid1d 10871	. . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐼 + 0) = 𝐼)
29	28	ad2antll 729	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + 0) = 𝐼)
30	25, 29	eqtrd 2794	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + (0 · 𝐿)) = 𝐼)
31	30	oveq1d 7166	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → ((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)) = (𝐼 mod (♯‘𝑊)))
32		zmodidfzoimp 13311	. . . . . . . 8 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐼 mod (♯‘𝑊)) = 𝐼)
33	32	ad2antll 729	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 mod (♯‘𝑊)) = 𝐼)
34	31, 33	eqtr2d 2795	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → 𝐼 = ((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)))
35	34	fveq2d 6663	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))
36		fveq1 6658	. . . . . . . . . . . . 13 ⊢ (𝑊 = (𝑊 cyclShift 𝐿) → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
37	36	eqcoms 2767	. . . . . . . . . . . 12 ⊢ ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
38	37	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
39	38	adantl 486	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
40		simprll 779	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → 𝑊 ∈ Word 𝑉)
41		simprlr 780	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → 𝐿 ∈ ℤ)
42		elfzo0 13120	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
43		nn0z 12037	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℤ)
44	43	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) → 𝐼 ∈ ℤ)
45		nn0z 12037	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
46		zmulcl 12063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑦 · 𝐿) ∈ ℤ)
47	45, 46	sylan 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐿 ∈ ℤ) → (𝑦 · 𝐿) ∈ ℤ)
48	47	ancoms 463	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → (𝑦 · 𝐿) ∈ ℤ)
49		zaddcl 12054	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℤ ∧ (𝑦 · 𝐿) ∈ ℤ) → (𝐼 + (𝑦 · 𝐿)) ∈ ℤ)
50	44, 48, 49	syl2an 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) ∧ (𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀)) → (𝐼 + (𝑦 · 𝐿)) ∈ ℤ)
51		simplr 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) ∧ (𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀)) → (♯‘𝑊) ∈ ℕ)
52	50, 51	jca 516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) ∧ (𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀)) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))
53	52	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
54	53	3adant3 1130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
55	42, 54	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
56	55	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
57	56	expd 420	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝐿 ∈ ℤ → (𝑦 ∈ ℕ₀ → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))))
58	57	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℤ → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑦 ∈ ℕ₀ → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))))
59	58	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑦 ∈ ℕ₀ → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))))
60	59	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑦 ∈ ℕ₀ → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
61	60	impcom 412	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))
62		zmodfzo 13304	. . . . . . . . . . . 12 ⊢ (((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ) → ((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) ∈ (0..^(♯‘𝑊)))
63	61, 62	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) ∈ (0..^(♯‘𝑊)))
64		cshwidxmod 14205	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ ∧ ((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊))))
65	40, 41, 63, 64	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊))))
66		nn0re 11936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
67		zre 12017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
68		nn0re 11936	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℝ)
69		nnrp 12434	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ⁺)
70		remulcl 10653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑦 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℝ)
71	70	ancoms 463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℝ)
72		readdcl 10651	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐼 ∈ ℝ ∧ (𝑦 · 𝐿) ∈ ℝ) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
73	71, 72	sylan2 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐼 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
74	73	ancoms 463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
75	74	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
76		simprll 779	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → 𝐿 ∈ ℝ)
77		simpl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → (♯‘𝑊) ∈ ℝ⁺)
78		modaddmod 13320	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐼 + (𝑦 · 𝐿)) ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ⁺) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = (((𝐼 + (𝑦 · 𝐿)) + 𝐿) mod (♯‘𝑊)))
79	75, 76, 77, 78	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = (((𝐼 + (𝑦 · 𝐿)) + 𝐿) mod (♯‘𝑊)))
80		recn 10658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℂ)
81	80	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → 𝐼 ∈ ℂ)
82	70	recnd 10700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑦 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℂ)
83	82	ancoms 463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℂ)
84	83	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℂ)
85		recn 10658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
86	85	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝐿 ∈ ℂ)
87	86	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → 𝐿 ∈ ℂ)
88	81, 84, 87	addassd 10694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 · 𝐿) + 𝐿)))
89		recn 10658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
90	89	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
91		1cnd 10667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
92	90, 91, 86	adddird 10697	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) · 𝐿) = ((𝑦 · 𝐿) + (1 · 𝐿)))
93	85	mulid2d 10690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐿 ∈ ℝ → (1 · 𝐿) = 𝐿)
94	93	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (1 · 𝐿) = 𝐿)
95	94	oveq2d 7167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 · 𝐿) + (1 · 𝐿)) = ((𝑦 · 𝐿) + 𝐿))
96	92, 95	eqtr2d 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 · 𝐿) + 𝐿) = ((𝑦 + 1) · 𝐿))
97	96	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝑦 · 𝐿) + 𝐿) = ((𝑦 + 1) · 𝐿))
98	97	oveq2d 7167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝐼 + ((𝑦 · 𝐿) + 𝐿)) = (𝐼 + ((𝑦 + 1) · 𝐿)))
99	88, 98	eqtrd 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 + 1) · 𝐿)))
100	99	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 + 1) · 𝐿)))
101	100	oveq1d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → (((𝐼 + (𝑦 · 𝐿)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))
102	79, 101	eqtrd 2794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))
103	102	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑊) ∈ ℝ⁺ → (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
104	69, 103	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑊) ∈ ℕ → (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
105	104	expd 420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑊) ∈ ℕ → ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐼 ∈ ℝ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
106	105	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((♯‘𝑊) ∈ ℕ → (𝐼 ∈ ℝ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
107	67, 68, 106	syl2an 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((♯‘𝑊) ∈ ℕ → (𝐼 ∈ ℝ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
108	107	com13 88	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℝ → ((♯‘𝑊) ∈ ℕ → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
109	66, 108	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ ℕ₀ → ((♯‘𝑊) ∈ ℕ → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
110	109	imp 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
111	110	3adant3 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
112	42, 111	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
113	112	expd 420	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐿 ∈ ℤ → (𝑦 ∈ ℕ₀ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
114	113	adantld 495	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (𝑦 ∈ ℕ₀ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
115	114	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (𝑦 ∈ ℕ₀ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
116	115	impcom 412	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑦 ∈ ℕ₀ → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
117	116	impcom 412	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊)) = ((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))
118	117	fveq2d 6663	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → (𝑊‘((((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)) + 𝐿) mod (♯‘𝑊))) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
119	39, 65, 118	3eqtrd 2798	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
120	119	eqeq2d 2770	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
121	120	biimpd 232	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
122	121	ex 417	. . . . . 6 ⊢ (𝑦 ∈ ℕ₀ → (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))))
123	122	a2d 29	. . . . 5 ⊢ (𝑦 ∈ ℕ₀ → ((((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)))) → (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))))
124	5, 10, 15, 20, 35, 123	nn0ind 12109	. . . 4 ⊢ (𝑗 ∈ ℕ₀ → (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))
125	124	com12 32	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑗 ∈ ℕ₀ → (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))
126	125	ralrimiv 3113	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → ∀𝑗 ∈ ℕ₀ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊))))
127	126	ex 417	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ∀𝑗 ∈ ℕ₀ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ∀wral 3071 class class class wbr 5033 ‘cfv 6336 (class class class)co 7151 ℂcc 10566 ℝcr 10567 0cc0 10568 1c1 10569 + caddc 10571 · cmul 10573 < clt 10706 ℕcn 11667 ℕ₀cn0 11927 ℤcz 12013 ℝ⁺crp 12423 ..^cfzo 13075 mod cmo 13279 ♯chash 13733 Word cword 13906 cyclShift ccsh 14190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-cnex 10624 ax-resscn 10625 ax-1cn 10626 ax-icn 10627 ax-addcl 10628 ax-addrcl 10629 ax-mulcl 10630 ax-mulrcl 10631 ax-mulcom 10632 ax-addass 10633 ax-mulass 10634 ax-distr 10635 ax-i2m1 10636 ax-1ne0 10637 ax-1rid 10638 ax-rnegex 10639 ax-rrecex 10640 ax-cnre 10641 ax-pre-lttri 10642 ax-pre-lttrn 10643 ax-pre-ltadd 10644 ax-pre-mulgt0 10645 ax-pre-sup 10646

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-int 4840 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-om 7581 df-1st 7694 df-2nd 7695 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-1o 8113 df-oadd 8117 df-er 8300 df-en 8529 df-dom 8530 df-sdom 8531 df-fin 8532 df-sup 8932 df-inf 8933 df-card 9394 df-pnf 10708 df-mnf 10709 df-xr 10710 df-ltxr 10711 df-le 10712 df-sub 10903 df-neg 10904 df-div 11329 df-nn 11668 df-2 11730 df-n0 11928 df-z 12014 df-uz 12276 df-rp 12424 df-fz 12933 df-fzo 13076 df-fl 13204 df-mod 13280 df-hash 13734 df-word 13907 df-concat 13963 df-substr 14043 df-pfx 14073 df-csh 14191

This theorem is referenced by: cshw1 14224 cshwsidrepsw 16478

W3C validator