2cshwcshw - Metamath Proof Explorer

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Theorem 2cshwcshw 14781

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 13620	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ ¬ 𝐾 ≤ 𝑚) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))
2	1	3exp 1118	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾 ≤ 𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))))
3	2	ad2antrr 723	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → 𝑌 ∈ Word 𝑉)
10	9	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉)
11		elfzelz 13506	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
12	11	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → 𝐾 ∈ ℤ)
13	12	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
14		elfz2 13496	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)))
15		zaddcl 12607	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 + 𝑁) ∈ ℤ)
16	15	adantrr 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝑚 + 𝑁) ∈ ℤ)
17		simprr 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈ ℤ)
18	16, 17	zsubcld 12676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)
19	18	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
20		elfzelz 13506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
23	22	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
26	25	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)
27		2cshw 14768	. . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . 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 12675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)
30	29	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
33	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
36	35	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾 ≤ 𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)
37		cshwsublen 14751	. . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . . 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 13597	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝐾 ≤ 𝑁))
41		nn0cn 12487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
42		nn0cn 12487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℂ)
43		nn0cn 12487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
44	42, 43	anim12i 612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ))
45		simprl 768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → 𝐾 ∈ ℂ)
46		addcl 11196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑚 + 𝑁) ∈ ℂ)
47	46	adantrl 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝑚 + 𝑁) ∈ ℂ)
48	45, 47	pncan3d 11579	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) = (𝑚 + 𝑁))
49	48	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = ((𝑚 + 𝑁) − 𝑁))
50		pncan 11471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑚 + 𝑁) − 𝑁) = 𝑚)
51	50	adantrl 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝑚 + 𝑁) − 𝑁) = 𝑚)
52	49, 51	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)
53	41, 44, 52	syl2an 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ ℕ₀ ∧ (𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)
54	53	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ℕ₀ → ((𝐾 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
55		elfznn0 13599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
60		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝑌) = 𝑁 → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁))
61	60	eqeq1d 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑌) = 𝑁 → (((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚 ↔ ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
62	61	imbi2d 340	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑌) = 𝑁 → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)))
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
72	71	adantl 481	. . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . 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 451	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 7420	. . . . . . . . . . . 12 ⊢ (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
82	81	rspceeqv 3633	. . . . . . . . . . 11 ⊢ ((((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
83	8, 80, 82	syl2anc 583	. . . . . . . . . 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 419	. . . . . . . . 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 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → (𝑌 cyclShift 𝑚) = (𝑌 cyclShift 0))
86	85	eqeq2d 2742	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑌 cyclShift 0)))
87		cshw0 14749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ∈ Word 𝑉 → (𝑌 cyclShift 0) = 𝑌)
88	87	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑌 cyclShift 0) = 𝑌)
89	88	eqeq2d 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) ↔ 𝑍 = 𝑌))
90		fznn0sub2 13613	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈ (0...𝑁))
91	90	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁 − 𝐾) ∈ (0...𝑁))
92		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑌) = 𝑁 → ((♯‘𝑌) − 𝐾) = (𝑁 − 𝐾))
93	92	eleq1d 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑌) = 𝑁 → (((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁 − 𝐾) ∈ (0...𝑁)))
94	93	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((♯‘𝑌) − 𝐾) ∈ (0...𝑁))
97		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((♯‘𝑌) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)))
98		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → 𝑌 ∈ Word 𝑉)
99		2cshwid 14769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑌 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
100	98, 11, 99	syl2an 595	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
101	97, 100	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
102	101	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → 𝑌 = (𝑋 cyclShift ((♯‘𝑌) − 𝐾)))
103		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = ((♯‘𝑌) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((♯‘𝑌) − 𝐾)))
104	103	rspceeqv 3633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ∧ 𝑌 = (𝑋 cyclShift ((♯‘𝑌) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
105	96, 102, 104	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
106	105	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
107		eqeq1 2735	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑍 = 𝑌 → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift 𝑛)))
108	107	rexbidv 3177	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑍 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
109	108	adantl 481	. . . . . . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 425	. . . . . . . . . 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 13619	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ 𝐾 ≤ 𝑚) → (𝑚 − 𝐾) ∈ (0...𝑁))
124	123	3exp 1118	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 ≤ 𝑚 → (𝑚 − 𝐾) ∈ (0...𝑁))))
125	124	ad2antrr 723	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉)
130	12	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
131		zsubcl 12609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 − 𝐾) ∈ ℤ)
132	131	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝑚 − 𝐾) ∈ ℤ))
135	134	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 − 𝐾) ∈ ℤ)
136		2cshw 14768	. . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . 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 12568	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
139	20	zcnd 12672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℂ)
140		pncan3 11473	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
141	138, 139, 140	syl2anr 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ (0...𝑁) ∧ 𝐾 ∈ ℤ) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
142	141	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚 − 𝐾)) = 𝑚))
145	144	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾 ≤ 𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + (𝑚 − 𝐾)) = 𝑚)
146	145	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift (𝑚 − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾)))
150	149	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = (𝑌 cyclShift 𝐾) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚 − 𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚 − 𝐾))))
151	150	adantl 481	. . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 7420	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift (𝑚 − 𝐾)))
162	161	rspceeqv 3633	. . . . . . . . . . 11 ⊢ (((𝑚 − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift (𝑚 − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
163	128, 160, 162	syl2anc 583	. . . . . . . . . 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 412	. . . . . . . . 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 3156	. . . . . . 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 412	. . . . . 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 412	. . . 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 1110	. 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 0cc0 11114 + caddc 11117 ≤ cle 11254 − cmin 11449 ℕ₀cn0 12477 ℤcz 12563 ...cfz 13489 ♯chash 14295 Word cword 14469 cyclShift ccsh 14743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-hash 14296 df-word 14470 df-concat 14526 df-substr 14596 df-pfx 14626 df-csh 14744

This theorem is referenced by: eleclclwwlknlem1 29581

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