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Theorem 2cshwcshw 14175
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
StepHypRef Expression
1 difelfznle 13009 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ ¬ 𝐾𝑚) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))
213exp 1111 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))))
32ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (¬ 𝐾𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁))))
43imp 407 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (¬ 𝐾𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
54adantr 481 . . . . . . . . . . . . . 14 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (¬ 𝐾𝑚 → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
65com12 32 . . . . . . . . . . . . 13 𝐾𝑚 → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
76adantl 482 . . . . . . . . . . . 12 ((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) → (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁)))
87imp 407 . . . . . . . . . . 11 (((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) ∧ ((((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 767 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → 𝑌 ∈ Word 𝑉)
109ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉)
11 elfzelz 12896 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℤ)
1211adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → 𝐾 ∈ ℤ)
1312ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
14 elfz2 12887 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐾 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾𝐾𝑁)))
15 zaddcl 12010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 + 𝑁) ∈ ℤ)
1615adantrr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝑚 + 𝑁) ∈ ℤ)
17 simprr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈ ℤ)
1816, 17zsubcld 12080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)
1918ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
20 elfzelz 12896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
2119, 20syl11 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
22213adant1 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
2322adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾𝐾𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
2414, 23sylbi 218 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
2524ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ))
2625imp 407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 + 𝑁) − 𝐾) ∈ ℤ)
27 2cshw 14163 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 ∈ Word 𝑉𝐾 ∈ ℤ ∧ ((𝑚 + 𝑁) − 𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))))
2810, 13, 26, 27syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))))
2917, 18zaddcld 12079 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑚 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)
3029ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
3130, 20syl11 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
32313adant1 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
3332adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (0 ≤ 𝐾𝐾𝑁)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
3414, 33sylbi 218 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
3534ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) → (𝑚 ∈ (0...𝑁) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ))
3635imp 407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ)
37 cshwsublen 14146 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 ∈ Word 𝑉 ∧ (𝐾 + ((𝑚 + 𝑁) − 𝐾)) ∈ ℤ) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))))
3810, 36, 37syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + ((𝑚 + 𝑁) − 𝐾))) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))))
3928, 38eqtrd 2853 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)) = (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))))
40 elfz2nn0 12986 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝑁 ∈ ℕ0𝐾𝑁))
41 nn0cn 11895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
42 nn0cn 11895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐾 ∈ ℕ0𝐾 ∈ ℂ)
43 nn0cn 11895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
4442, 43anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ))
45 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → 𝐾 ∈ ℂ)
46 addcl 10607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑚 + 𝑁) ∈ ℂ)
4746adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝑚 + 𝑁) ∈ ℂ)
4845, 47pncan3d 10988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐾 + ((𝑚 + 𝑁) − 𝐾)) = (𝑚 + 𝑁))
4948oveq1d 7160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = ((𝑚 + 𝑁) − 𝑁))
50 pncan 10880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑚 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑚 + 𝑁) − 𝑁) = 𝑚)
5150adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝑚 + 𝑁) − 𝑁) = 𝑚)
5249, 51eqtrd 2853 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑚 ∈ ℂ ∧ (𝐾 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)
5341, 44, 52syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑚 ∈ ℕ0 ∧ (𝐾 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)
5453ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ ℕ0 → ((𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
55 elfznn0 12988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ0)
5654, 55syl11 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
57563adant3 1124 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐾 ∈ ℕ0𝑁 ∈ ℕ0𝐾𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
5840, 57sylbi 218 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
5958adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
60 oveq2 7153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((♯‘𝑌) = 𝑁 → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁))
6160eqeq1d 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝑌) = 𝑁 → (((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚 ↔ ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚))
6261imbi2d 342 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑌) = 𝑁 → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)))
6362adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → ((𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚) ↔ (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − 𝑁) = 𝑚)))
6463adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
6559, 64mpbird 258 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚))
6665adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) → (𝑚 ∈ (0...𝑁) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚))
6766imp 407 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌)) = 𝑚)
6867oveq2d 7161 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift ((𝐾 + ((𝑚 + 𝑁) − 𝐾)) − (♯‘𝑌))) = (𝑌 cyclShift 𝑚))
6939, 68eqtr2d 2854 . . . . . . . . . . . . . . . . . . . 20 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
7069adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
71 oveq1 7152 . . . . . . . . . . . . . . . . . . . 20 (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
7271adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((𝑚 + 𝑁) − 𝐾)))
7370, 72eqtr4d 2856 . . . . . . . . . . . . . . . . . 18 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
7473exp41 435 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))))
7574com24 95 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))))
7675imp41 426 . . . . . . . . . . . . . . 15 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
7776eqeq2d 2829 . . . . . . . . . . . . . 14 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))
7877biimpd 230 . . . . . . . . . . . . 13 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ (¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚)) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))
7978impancom 452 . . . . . . . . . . . 12 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → ((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) → 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))))
8079impcom 408 . . . . . . . . . . 11 (((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) ∧ ((((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 7153 . . . . . . . . . . . 12 (𝑛 = ((𝑚 + 𝑁) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾)))
8281rspceeqv 3635 . . . . . . . . . . 11 ((((𝑚 + 𝑁) − 𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift ((𝑚 + 𝑁) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
838, 80, 82syl2anc 584 . . . . . . . . . 10 (((¬ 𝑚 = 0 ∧ ¬ 𝐾𝑚) ∧ ((((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))
8483exp31 420 . . . . . . . . 9 𝑚 = 0 → (¬ 𝐾𝑚 → (((((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 7153 . . . . . . . . . . . . . . 15 (𝑚 = 0 → (𝑌 cyclShift 𝑚) = (𝑌 cyclShift 0))
8685eqeq2d 2829 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑌 cyclShift 0)))
87 cshw0 14144 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 ∈ Word 𝑉 → (𝑌 cyclShift 0) = 𝑌)
8887adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑌 cyclShift 0) = 𝑌)
8988eqeq2d 2829 . . . . . . . . . . . . . . . . . . 19 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) ↔ 𝑍 = 𝑌))
90 fznn0sub2 13002 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐾 ∈ (0...𝑁) → (𝑁𝐾) ∈ (0...𝑁))
9190adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (𝑁𝐾) ∈ (0...𝑁))
92 oveq1 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝑌) = 𝑁 → ((♯‘𝑌) − 𝐾) = (𝑁𝐾))
9392eleq1d 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑌) = 𝑁 → (((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁𝐾) ∈ (0...𝑁)))
9493ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → (((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ↔ (𝑁𝐾) ∈ (0...𝑁)))
9591, 94mpbird 258 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑌) − 𝐾) ∈ (0...𝑁))
9695adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((♯‘𝑌) − 𝐾) ∈ (0...𝑁))
97 oveq1 7152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift ((♯‘𝑌) − 𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)))
98 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → 𝑌 ∈ Word 𝑉)
99 2cshwid 14164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑌 ∈ Word 𝑉𝐾 ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
10098, 11, 99syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
10197, 100sylan9eqr 2875 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑋 cyclShift ((♯‘𝑌) − 𝐾)) = 𝑌)
102101eqcomd 2824 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → 𝑌 = (𝑋 cyclShift ((♯‘𝑌) − 𝐾)))
103 oveq2 7153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((♯‘𝑌) − 𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift ((♯‘𝑌) − 𝐾)))
104103rspceeqv 3635 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((♯‘𝑌) − 𝐾) ∈ (0...𝑁) ∧ 𝑌 = (𝑋 cyclShift ((♯‘𝑌) − 𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
10596, 102, 104syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
106105adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
107 eqeq1 2822 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑍 = 𝑌 → (𝑍 = (𝑋 cyclShift 𝑛) ↔ 𝑌 = (𝑋 cyclShift 𝑛)))
108107rexbidv 3294 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑍 = 𝑌 → (∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
109108adantl 482 . . . . . . . . . . . . . . . . . . . . . 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)))
110106, 109mpbird 258 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) ∧ 𝐾 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑍 = 𝑌) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
111110exp41 435 . . . . . . . . . . . . . . . . . . . 20 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝐾 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = 𝑌 → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
112111com24 95 . . . . . . . . . . . . . . . . . . 19 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = 𝑌 → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
11389, 112sylbid 241 . . . . . . . . . . . . . . . . . 18 ((𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁) → (𝑍 = (𝑌 cyclShift 0) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝐾 ∈ (0...𝑁) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))))
114113com24 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)))))
115114impcom 408 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 0) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))))
116115com13 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))))
117116a1d 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)))))
11886, 117syl6bi 254 . . . . . . . . . . . . 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))))))
119118com24 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))))))
120119com15 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))))))
121120imp41 426 . . . . . . . . . 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)))
122121com12 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 13008 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (0...𝑁) ∧ 𝑚 ∈ (0...𝑁) ∧ 𝐾𝑚) → (𝑚𝐾) ∈ (0...𝑁))
1241233exp 1111 . . . . . . . . . . . . . . 15 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾𝑚 → (𝑚𝐾) ∈ (0...𝑁))))
125124ad2antrr 722 . . . . . . . . . . . . . 14 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑚 ∈ (0...𝑁) → (𝐾𝑚 → (𝑚𝐾) ∈ (0...𝑁))))
126125imp 407 . . . . . . . . . . . . 13 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾𝑚 → (𝑚𝐾) ∈ (0...𝑁)))
127126adantr 481 . . . . . . . . . . . 12 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝐾𝑚 → (𝑚𝐾) ∈ (0...𝑁)))
128127impcom 408 . . . . . . . . . . 11 ((𝐾𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → (𝑚𝐾) ∈ (0...𝑁))
1299ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝑌 ∈ Word 𝑉)
13012ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → 𝐾 ∈ ℤ)
131 zsubcl 12012 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑚𝐾) ∈ ℤ)
132131ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ ℤ → (𝐾 ∈ ℤ → (𝑚𝐾) ∈ ℤ))
13320, 11, 132syl2imc 41 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝑚𝐾) ∈ ℤ))
134133ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) → (𝑚 ∈ (0...𝑁) → (𝑚𝐾) ∈ ℤ))
135134imp 407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚𝐾) ∈ ℤ)
136 2cshw 14163 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 ∈ Word 𝑉𝐾 ∈ ℤ ∧ (𝑚𝐾) ∈ ℤ) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚𝐾))))
137129, 130, 135, 136syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾)) = (𝑌 cyclShift (𝐾 + (𝑚𝐾))))
138 zcn 11974 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
13920zcnd 12076 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℂ)
140 pncan3 10882 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐾 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐾 + (𝑚𝐾)) = 𝑚)
141138, 139, 140syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ (0...𝑁) ∧ 𝐾 ∈ ℤ) → (𝐾 + (𝑚𝐾)) = 𝑚)
142141ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (0...𝑁) → (𝐾 ∈ ℤ → (𝐾 + (𝑚𝐾)) = 𝑚))
14311, 142syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐾 ∈ (0...𝑁) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚𝐾)) = 𝑚))
144143ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) → (𝑚 ∈ (0...𝑁) → (𝐾 + (𝑚𝐾)) = 𝑚))
145144imp 407 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾 + (𝑚𝐾)) = 𝑚)
146145oveq2d 7161 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift (𝐾 + (𝑚𝐾))) = (𝑌 cyclShift 𝑚))
147137, 146eqtr2d 2854 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾)))
148147adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾)))
149 oveq1 7152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑋 = (𝑌 cyclShift 𝐾) → (𝑋 cyclShift (𝑚𝐾)) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾)))
150149eqeq2d 2829 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 = (𝑌 cyclShift 𝐾) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾))))
151150adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → ((𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚𝐾)) ↔ (𝑌 cyclShift 𝑚) = ((𝑌 cyclShift 𝐾) cyclShift (𝑚𝐾))))
152148, 151mpbird 258 . . . . . . . . . . . . . . . . . . 19 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑌 cyclShift 𝑚) = (𝑋 cyclShift (𝑚𝐾)))
153152eqeq2d 2829 . . . . . . . . . . . . . . . . . 18 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑍 = (𝑌 cyclShift 𝑚) ↔ 𝑍 = (𝑋 cyclShift (𝑚𝐾))))
154153biimpd 230 . . . . . . . . . . . . . . . . 17 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝐾𝑚) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚𝐾))))
155154exp41 435 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝐾𝑚 → (𝑚 ∈ (0...𝑁) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚𝐾)))))))
156155com24 95 . . . . . . . . . . . . . . 15 ((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) → (𝑋 = (𝑌 cyclShift 𝐾) → (𝑚 ∈ (0...𝑁) → (𝐾𝑚 → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚𝐾)))))))
157156imp31 418 . . . . . . . . . . . . . 14 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝐾𝑚 → (𝑍 = (𝑌 cyclShift 𝑚) → 𝑍 = (𝑋 cyclShift (𝑚𝐾)))))
158157com23 86 . . . . . . . . . . . . 13 ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) → (𝑍 = (𝑌 cyclShift 𝑚) → (𝐾𝑚𝑍 = (𝑋 cyclShift (𝑚𝐾)))))
159158imp 407 . . . . . . . . . . . 12 (((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚)) → (𝐾𝑚𝑍 = (𝑋 cyclShift (𝑚𝐾))))
160159impcom 408 . . . . . . . . . . 11 ((𝐾𝑚 ∧ ((((𝐾 ∈ (0...𝑁) ∧ (𝑌 ∈ Word 𝑉 ∧ (♯‘𝑌) = 𝑁)) ∧ 𝑋 = (𝑌 cyclShift 𝐾)) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑍 = (𝑌 cyclShift 𝑚))) → 𝑍 = (𝑋 cyclShift (𝑚𝐾)))
161 oveq2 7153 . . . . . . . . . . . 12 (𝑛 = (𝑚𝐾) → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift (𝑚𝐾)))
162161rspceeqv 3635 . . . . . . . . . . 11 (((𝑚𝐾) ∈ (0...𝑁) ∧ 𝑍 = (𝑋 cyclShift (𝑚𝐾))) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))
163128, 160, 162syl2anc 584 . . . . . . . . . 10 ((𝐾𝑚 ∧ ((((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))
164163ex 413 . . . . . . . . 9 (𝐾𝑚 → (((((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)))
16584, 122, 164pm2.61ii 184 . . . . . . . 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))
166165rexlimdva2 3284 . . . . . . 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)))
167166ex 413 . . . . . 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))))
168167com23 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))))
169168ex 413 . . . 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)))))
170169com24 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)))))
1711703imp 1103 . 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)))
172171com12 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525   + caddc 10528  cle 10664  cmin 10858  0cn0 11885  cz 11969  ...cfz 12880  chash 13678  Word cword 13849   cyclShift ccsh 14138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-hash 13679  df-word 13850  df-concat 13911  df-substr 13991  df-pfx 14021  df-csh 14139
This theorem is referenced by:  eleclclwwlknlem1  27766
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