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Theorem cshweqrep 14767

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 7412	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 · 𝐿) = (0 · 𝐿))
2	1	oveq2d 7421	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (0 · 𝐿)))
3	2	fvoveq1d 7427	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))
4	3	eqeq2d 2743	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)))))
5	4	imbi2d 340	. . . . 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 7412	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 · 𝐿) = (𝑦 · 𝐿))
7	6	oveq2d 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (𝑦 · 𝐿)))
8	7	fvoveq1d 7427	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
9	8	eqeq2d 2743	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊)))))
10	9	imbi2d 340	. . . . 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 7412	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝐿) = ((𝑦 + 1) · 𝐿))
12	11	oveq2d 7421	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + ((𝑦 + 1) · 𝐿)))
13	12	fvoveq1d 7427	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊))))
14	13	eqeq2d 2743	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + ((𝑦 + 1) · 𝐿)) mod (♯‘𝑊)))))
15	14	imbi2d 340	. . . . 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 7412	. . . . . . . . 9 ⊢ (𝑥 = 𝑗 → (𝑥 · 𝐿) = (𝑗 · 𝐿))
17	16	oveq2d 7421	. . . . . . . 8 ⊢ (𝑥 = 𝑗 → (𝐼 + (𝑥 · 𝐿)) = (𝐼 + (𝑗 · 𝐿)))
18	17	fvoveq1d 7427	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊))))
19	18	eqeq2d 2743	. . . . . 6 ⊢ (𝑥 = 𝑗 → ((𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑥 · 𝐿)) mod (♯‘𝑊))) ↔ (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))
20	19	imbi2d 340	. . . . 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 12559	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
22	21	mul02d 11408	. . . . . . . . . . . 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 7421	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + (0 · 𝐿)) = (𝐼 + 0))
26		elfzoelz 13628	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → 𝐼 ∈ ℤ)
27	26	zcnd 12663	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → 𝐼 ∈ ℂ)
28	27	addridd 11410	. . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐼 + 0) = 𝐼)
29	28	ad2antll 727	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + 0) = 𝐼)
30	25, 29	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 + (0 · 𝐿)) = 𝐼)
31	30	oveq1d 7420	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → ((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)) = (𝐼 mod (♯‘𝑊)))
32		zmodidfzoimp 13862	. . . . . . . 8 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (𝐼 mod (♯‘𝑊)) = 𝐼)
33	32	ad2antll 727	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝐼 mod (♯‘𝑊)) = 𝐼)
34	31, 33	eqtr2d 2773	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → 𝐼 = ((𝐼 + (0 · 𝐿)) mod (♯‘𝑊)))
35	34	fveq2d 6892	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊)))) → (𝑊‘𝐼) = (𝑊‘((𝐼 + (0 · 𝐿)) mod (♯‘𝑊))))
36		fveq1 6887	. . . . . . . . . . . . 13 ⊢ (𝑊 = (𝑊 cyclShift 𝐿) → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
37	36	eqcoms 2740	. . . . . . . . . . . 12 ⊢ ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))) = ((𝑊 cyclShift 𝐿)‘((𝐼 + (𝑦 · 𝐿)) mod (♯‘𝑊))))
38	37	ad2antrl 726	. . . . . . . . . . 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 777	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → 𝑊 ∈ Word 𝑉)
41		simprlr 778	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ₀ ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) ∧ ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))))) → 𝐿 ∈ ℤ)
42		elfzo0 13669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
43		nn0z 12579	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℤ)
44	43	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ) → 𝐼 ∈ ℤ)
45		nn0z 12579	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
46		zmulcl 12607	. . . . . . . . . . . . . . . . . . . . . . . . 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 12598	. . . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . 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 13855	. . . . . . . . . . . 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 14749	. . . . . . . . . . 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 1371	. . . . . . . . . 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 12477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ ℕ₀ → 𝐼 ∈ ℝ)
67		zre 12558	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
68		nn0re 12477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℝ)
69		nnrp 12981	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ⁺)
70		remulcl 11191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑦 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℝ)
71	70	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · 𝐿) ∈ ℝ)
72		readdcl 11189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13871	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℂ)
81	80	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → 𝐼 ∈ ℂ)
82	70	recnd 11238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → ((𝐼 + (𝑦 · 𝐿)) + 𝐿) = (𝐼 + ((𝑦 · 𝐿) + 𝐿)))
89		recn 11196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
90	89	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
91		1cnd 11205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
92	90, 91, 86	adddird 11235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) · 𝐿) = ((𝑦 · 𝐿) + (1 · 𝐿)))
93	85	mullidd 11228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐿 ∈ ℝ → (1 · 𝐿) = 𝐿)
94	93	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (1 · 𝐿) = 𝐿)
95	94	oveq2d 7421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 · 𝐿) + (1 · 𝐿)) = ((𝑦 · 𝐿) + 𝐿))
96	92, 95	eqtr2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝐼 ∈ ℝ) → (𝐼 + ((𝑦 · 𝐿) + 𝐿)) = (𝐼 + ((𝑦 + 1) · 𝐿)))
99	88, 98	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . 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 2776	. . . . . . . . 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 2743	. . . . . . . 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 12653	. . . 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 3145	. 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℝ⁺crp 12970 ..^cfzo 13623 mod cmo 13830 ♯chash 14286 Word cword 14460 cyclShift ccsh 14734

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-hash 14287 df-word 14461 df-concat 14517 df-substr 14587 df-pfx 14617 df-csh 14735

This theorem is referenced by: cshw1 14768 cshwsidrepsw 17023

W3C validator