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Theorem cshweqrep 14534

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 7282	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 · 𝐿) = (0 · 𝐿))
2	1	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (0 · 𝐿)))
3	2	fvoveq1d 7297	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))
4	3	eqeq2d 2749	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)))))
5	4	imbi2d 341	. . . . 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 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐿) = (𝑦 · 𝐿))
7	6	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (𝑦 · 𝐿)))
8	7	fvoveq1d 7297	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
9	8	eqeq2d 2749	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)))))
10	9	imbi2d 341	. . . . 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 7282	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝐿) = ((𝑦 + 1) · 𝐿))
12	11	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + ((𝑦 + 1) · 𝐿)))
13	12	fvoveq1d 7297	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
14	13	eqeq2d 2749	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
15	14	imbi2d 341	. . . . 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 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (𝑥 · 𝐿) = (𝑗 · 𝐿))
17	16	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = 𝑗 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (𝑗 · 𝐿)))
18	17	fvoveq1d 7297	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊))))
19	18	eqeq2d 2749	. . . . . 6 ⊢ (𝑥 = 𝑗 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))
20	19	imbi2d 341	. . . . 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 12324	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
22	21	mul02d 11173	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ ℤ → (0 · 𝐿) = 0)
23	22	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (0 · 𝐿) = 0)
24	23	adantr 481	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (0 · 𝐿) = 0)
25	24	oveq2d 7291	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + (0 · 𝐿)) = (𝐼 + 0))
26		elfzoelz 13387	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → 𝐼 ∈ ℤ)
27	26	zcnd 12427	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → 𝐼 ∈ ℂ)
28	27	addid1d 11175	. . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐼 + 0) = 𝐼)
29	28	ad2antll 726	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + 0) = 𝐼)
30	25, 29	eqtrd 2778	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + (0 · 𝐿)) = 𝐼)
31	30	oveq1d 7290	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → ((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)) = (𝐼 mod (♯‘𝑊)))
32		zmodidfzoimp 13621	. . . . . . . 8 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐼 mod (♯‘𝑊)) = 𝐼)
33	32	ad2antll 726	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 mod (♯‘𝑊)) = 𝐼)
34	31, 33	eqtr2d 2779	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → 𝐼 = ((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)))
35	34	fveq2d 6778	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))
36		fveq1 6773	. . . . . . . . . . . . 13 ⊢ (𝑊 = (𝑊 cyclShift 𝐿) → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
37	36	eqcoms 2746	. . . . . . . . . . . 12 ⊢ ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
38	37	ad2antrl 725	. . . . . . . . . . 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 482	. . . . . . . . . 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 776	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → 𝑊 ∈ Word 𝑉)
41		simprlr 777	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → 𝐿 ∈ ℤ)
42		elfzo0 13428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
43		nn0z 12343	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℤ)
44	43	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) → 𝐼 ∈ ℤ)
45		nn0z 12343	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
46		zmulcl 12369	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑦 · 𝐿) ∈ ℤ)
47	45, 46	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐿 ∈ ℤ) → (𝑦 · 𝐿) ∈ ℤ)
48	47	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → (𝑦 · 𝐿) ∈ ℤ)
49		zaddcl 12360	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℤ ∧ (𝑦 · 𝐿) ∈ ℤ) → (𝐼 + (𝑦 · 𝐿)) ∈ ℤ)
50	44, 48, 49	syl2an 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) ∧ (𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀)) → (𝐼 + (𝑦 · 𝐿)) ∈ ℤ)
51		simplr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) ∧ (𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀)) → (♯‘𝑊) ∈ ℕ)
52	50, 51	jca 512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) ∧ (𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀)) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))
53	52	ex 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
54	53	3adant3 1131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
55	42, 54	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
56	55	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℕ₀) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
57	56	expd 416	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑦 ∈ ℕ₀ → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))))
60	59	imp 407	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑦 ∈ ℕ₀ → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ)))
61	60	impcom 408	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → ((𝐼 + (𝑦 · 𝐿)) ∈ ℤ ∧ (♯‘𝑊) ∈ ℕ))
62		zmodfzo 13614	. . . . . . . . . . . 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 14516	. . . . . . . . . . 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 1370	. . . . . . . . . 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 12242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
67		zre 12323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
68		nn0re 12242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℝ)
69		nnrp 12741	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ⁺)
70		remulcl 10956	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑦 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℝ)
71	70	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℝ)
72		readdcl 10954	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐼 ∈ ℝ ∧ (𝑦 · 𝐿) ∈ ℝ) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
73	71, 72	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐼 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
74	73	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
75	74	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → (𝐼 + (𝑦 · 𝐿)) ∈ ℝ)
76		simprll 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → 𝐿 ∈ ℝ)
77		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → (♯‘𝑊) ∈ ℝ⁺)
78		modaddmod 13630	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 10961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℂ)
81	80	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → 𝐼 ∈ ℂ)
82	70	recnd 11003	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑦 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℂ)
83	82	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℂ)
84	83	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℂ)
85		recn 10961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
86	85	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝐿 ∈ ℂ)
87	86	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → 𝐿 ∈ ℂ)
88	81, 84, 87	addassd 10997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 · 𝐿) + 𝐿)))
89		recn 10961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
90	89	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
91		1cnd 10970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
92	90, 91, 86	adddird 11000	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) · 𝐿) = ((𝑦 · 𝐿) + (1 · 𝐿)))
93	85	mulid2d 10993	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐿 ∈ ℝ → (1 · 𝐿) = 𝐿)
94	93	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (1 · 𝐿) = 𝐿)
95	94	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 · 𝐿) + (1 · 𝐿)) = ((𝑦 · 𝐿) + 𝐿))
96	92, 95	eqtr2d 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 · 𝐿) + 𝐿) = ((𝑦 + 1) · 𝐿))
97	96	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝑦 · 𝐿) + 𝐿) = ((𝑦 + 1) · 𝐿))
98	97	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝐼 + ((𝑦 · 𝐿) + 𝐿)) = (𝐼 + ((𝑦 + 1) · 𝐿)))
99	88, 98	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 + 1) · 𝐿)))
100	99	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑊) ∈ ℝ⁺ ∧ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ)) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 + 1) · 𝐿)))
101	100	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 408	. . . . . . . . . . . 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 408	. . . . . . . . . . 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 6778	. . . . . . . . . 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 2782	. . . . . . . . 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 2749	. . . . . . . 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 228	. . . . . . 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 413	. . . . . 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 12415	. . . 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 3102	. 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 413	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℝ⁺crp 12730 ..^cfzo 13382 mod cmo 13589 ♯chash 14044 Word cword 14217 cyclShift ccsh 14501

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-hash 14045 df-word 14218 df-concat 14274 df-substr 14354 df-pfx 14384 df-csh 14502

This theorem is referenced by: cshw1 14535 cshwsidrepsw 16795

W3C validator