2cshwcshw - Metamath Proof Explorer

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Theorem 2cshwcshw 14583

Description: If a word is a cyclically shifted word, and a second word is the result of cyclically shifting the same word, then the second word is the result of cyclically shifting the first word. (Contributed by AV, 11-May-2018.) (Revised by AV, 12-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)

**Assertion**
Ref	Expression
2cshwcshw	⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))

Distinct variable groups: 𝑚,𝑛,𝐾 𝑚,𝑁,𝑛 𝑚,𝑉,𝑛 𝑚,𝑋,𝑛 𝑚,𝑌,𝑛 𝑚,𝑍,𝑛

**Proof of Theorem 2cshwcshw**
Step	Hyp	Ref	Expression
1		difelfznle 13416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ ¬ 𝐾 ≤ 𝑚) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))
2	1	3exp 1119	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))))
3	2	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))))
4	3	imp 408	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
5	4	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
6	5	com12 32	. . . . . . . . . . . . 13 ⊢ (¬ 𝐾 ≤ 𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
7	6	adantl 483	. . . . . . . . . . . 12 ⊢ ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
8	7	imp 408	. . . . . . . . . . 11 ⊢ (((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))
9		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → 𝑌 ∈ Word 𝑉)
10	9	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉)
11		elfzelz 13302	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
12	11	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → 𝐾 ∈ ℤ)
13	12	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
14		elfz2 13292	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
15		zaddcl 12406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 + 𝑁) ∈ ℤ)
16	15	adantrr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝑚 + 𝑁) ∈ ℤ)
17		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈ ℤ)
18	16, 17	zsubcld 12477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)
19	18	ex 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
20		elfzelz 13302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
21	19, 20	syl11 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
22	21	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
23	22	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
24	14, 23	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
25	24	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
26	25	imp 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)
27		2cshw 14571	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))))
28	10, 13, 26, 27	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))))
29	17, 18	zaddcld 12476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)
30	29	ex 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
31	30, 20	syl11 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
32	31	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
33	32	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
34	14, 33	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
35	34	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
36	35	imp 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)
37		cshwsublen 14554	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))))
38	10, 36, 37	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))))
39	28, 38	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))))
40		elfz2nn0 13393	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝐾 ≤ 𝑁))
41		nn0cn 12289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
42		nn0cn 12289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℂ)
43		nn0cn 12289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
44	42, 43	anim12i 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ))
45		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → 𝐾 ∈ ℂ)
46		addcl 10999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑚 + 𝑁) ∈ ℂ)
47	46	adantrl 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝑚 + 𝑁) ∈ ℂ)
48	45, 47	pncan3d 11381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) = (𝑚 + 𝑁))
49	48	oveq1d 7322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = ((𝑚 + 𝑁) − 𝑁))
50		pncan 11273	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑚 + 𝑁) − 𝑁) = 𝑚)
51	50	adantrl 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝑚 + 𝑁) − 𝑁) = 𝑚)
52	49, 51	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)
53	41, 44, 52	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)
54	53	ex 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ℕ₀ → ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
55		elfznn0 13395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ₀)
56	54, 55	syl11 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
57	56	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝐾 ≤ 𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
58	40, 57	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
59	58	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
60		oveq2 7315	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝑌) = 𝑁 → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁))
61	60	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑌) = 𝑁 → (((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚 ↔ ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
62	61	imbi2d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑌) = 𝑁 → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)))
63	62	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)))
64	63	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)))
65	59, 64	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚))
66	65	adantr 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚))
67	66	imp 408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚)
68	67	oveq2d 7323	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))) = (𝑌 cyclShift 𝑚))
69	39, 68	eqtr2d 2777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
70	69	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
71		oveq1 7314	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
72	71	adantl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
73	70, 72	eqtr4d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
74	73	exp41 436	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))))
75	74	com24 95	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))))
76	75	imp41 427	. . . . . . . . . . . . . . 15 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
77	76	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))
78	77	biimpd 228	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))
79	78	impancom 453	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))
80	79	impcom 409	. . . . . . . . . . 11 ⊢ (((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
81		oveq2 7315	. . . . . . . . . . . 12 ⊢ (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
82	81	rspceeqv 3580	. . . . . . . . . . 11 ⊢ ((((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
83	8, 80, 82	syl2anc 585	. . . . . . . . . 10 ⊢ (((¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚) ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
84	83	exp31 421	. . . . . . . . 9 ⊢ (¬ 𝑚 = 0 → (¬ 𝐾 ≤ 𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))
85		oveq2 7315	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑌 cyclShift 𝑚) = (𝑌 cyclShift 0))
86	85	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑌 cyclShift 0)))
87		cshw0 14552	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ∈ Word 𝑉 → (𝑌 cyclShift 0) = 𝑌)
88	87	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑌 cyclShift 0) = 𝑌)
89	88	eqeq2d 2747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) ↔ 𝑍 = 𝑌))
90		fznn0sub2 13409	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
91	90	adantl 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁))
92		oveq1 7314	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑌) = 𝑁 → ((♯‘𝑌) − 𝐾) = (𝑁 − 𝐾))
93	92	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑌) = 𝑁 → (((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁)))
94	93	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁)))
95	91, 94	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑌) − 𝐾) ∈ (0...𝑁))
96	95	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((♯‘𝑌) − 𝐾) ∈ (0...𝑁))
97		oveq1 7314	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((♯‘𝑌) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)))
98		simpl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → 𝑌 ∈ Word 𝑉)
99		2cshwid 14572	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
100	98, 11, 99	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
101	97, 100	sylan9eqr 2798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
102	101	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → 𝑌 = (𝑋 cyclShift ((♯‘𝑌) − 𝐾)))
103		oveq2 7315	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = ((♯‘𝑌) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((♯‘𝑌) − 𝐾)))
104	103	rspceeqv 3580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ∧ 𝑌 = (𝑋 cyclShift ((♯‘𝑌) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
105	96, 102, 104	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
106	105	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
107		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑍 = 𝑌 → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift 𝑛)))
108	107	rexbidv 3172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑍 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
109	108	adantl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
110	106, 109	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
111	110	exp41 436	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = 𝑌 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
112	111	com24 95	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = 𝑌 → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
113	89, 112	sylbid 239	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
114	113	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
115	114	impcom 409	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))
116	115	com13 88	. . . . . . . . . . . . . . 15 ⊢ (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))
117	116	a1d 25	. . . . . . . . . . . . . 14 ⊢ (𝑍 = (𝑌 cyclShift 0) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
118	86, 117	syl6bi 253	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))))
119	118	com24 95	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝑍 = (𝑌 cyclShift 𝑚) → ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))))
120	119	com15 101	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝑍 = (𝑌 cyclShift 𝑚) → (𝑚 = 0 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))))
121	120	imp41 427	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝑚 = 0 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
122	121	com12 32	. . . . . . . . 9 ⊢ (𝑚 = 0 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
123		difelfzle 13415	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ 𝐾 ≤ 𝑚) → (𝑚 − 𝐾) ∈ (0...𝑁))
124	123	3exp 1119	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁))))
125	124	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁))))
126	125	imp 408	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁)))
127	126	adantr 482	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁)))
128	127	impcom 409	. . . . . . . . . . 11 ⊢ ((𝐾 ≤ 𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → (𝑚 − 𝐾) ∈ (0...𝑁))
129	9	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉)
130	12	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
131		zsubcl 12408	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 − 𝐾) ∈ ℤ)
132	131	ex 414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℤ → (𝐾 ∈ ℤ → (𝑚 − 𝐾) ∈ ℤ))
133	20, 11, 132	syl2imc 41	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ))
134	133	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ))
135	134	imp 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 − 𝐾) ∈ ℤ)
136		2cshw 14571	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ (𝑚 − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾))))
137	129, 130, 135, 136	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾))))
138		zcn 12370	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
139	20	zcnd 12473	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℂ)
140		pncan3 11275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
141	138, 139, 140	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ (0...𝑁) ∧ 𝐾 ∈ ℤ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
142	141	ex 414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ (0...𝑁) → (𝐾 ∈ ℤ → (𝐾 + (𝑚 − 𝐾)) = 𝑚))
143	11, 142	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚))
144	143	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚))
145	144	imp 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
146	145	oveq2d 7323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + (𝑚 − 𝐾))) = (𝑌 cyclShift 𝑚))
147	137, 146	eqtr2d 2777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))
148	147	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))
149		oveq1 7314	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift (𝑚 − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))
150	149	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))))
151	150	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))))
152	148, 151	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)))
153	152	eqeq2d 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))
154	153	biimpd 228	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))
155	154	exp41 436	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝐾 ≤ 𝑚 → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))))))
156	155	com24 95	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))))))
157	156	imp31 419	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 ≤ 𝑚 → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))))
158	157	com23 86	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑍 = (𝑌 cyclShift 𝑚) → (𝐾 ≤ 𝑚 → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))))
159	158	imp 408	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝐾 ≤ 𝑚 → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))))
160	159	impcom 409	. . . . . . . . . . 11 ⊢ ((𝐾 ≤ 𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾)))
161		oveq2 7315	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift (𝑚 − 𝐾)))
162	161	rspceeqv 3580	. . . . . . . . . . 11 ⊢ (((𝑚 − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
163	128, 160, 162	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐾 ≤ 𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
164	163	ex 414	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
165	84, 122, 164	pm2.61ii 183	. . . . . . . 8 ⊢ (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
166	165	rexlimdva2 3151	. . . . . . 7 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
167	166	ex 414	. . . . . 6 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))
168	167	com23 86	. . . . 5 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → (𝑋 = (𝑌 cyclShift 𝐾) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))
169	168	ex 414	. . . 4 ⊢ (𝐾 ∈ (0...𝑁) → ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → (𝑋 = (𝑌 cyclShift 𝐾) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
170	169	com24 95	. . 3 ⊢ (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚) → ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
171	170	3imp 1111	. 2 ⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
172	171	com12 32	1 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 ∃wrex 3071 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 ℂcc 10915 0cc0 10917 + caddc 10920 ≤ cle 11056 − cmin 11251 ℕ₀cn0 12279 ℤcz 12365 ...cfz 13285 ♯chash 14090 Word cword 14262 cyclShift ccsh 14546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9245 df-inf 9246 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-n0 12280 df-z 12366 df-uz 12629 df-rp 12777 df-fz 13286 df-fzo 13429 df-fl 13558 df-mod 13636 df-hash 14091 df-word 14263 df-concat 14319 df-substr 14399 df-pfx 14429 df-csh 14547

This theorem is referenced by: eleclclwwlknlem1 28469

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