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Theorem cshwcsh2id 13620
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 26979 and erclwwlkntr 27035. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
Hypotheses
Ref Expression
cshwcsh2id.1 (𝜑𝑧 ∈ Word 𝑉)
cshwcsh2id.2 (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
Assertion
Ref Expression
cshwcsh2id (𝜑 → (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
Distinct variable group:   𝑘,𝑚,𝑛,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑘,𝑚,𝑛)   𝑉(𝑥,𝑦,𝑧,𝑘,𝑚,𝑛)

Proof of Theorem cshwcsh2id
StepHypRef Expression
1 oveq1 6697 . . . . . . . . 9 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
21eqeq2d 2661 . . . . . . . 8 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
32anbi2d 740 . . . . . . 7 (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
43adantr 480 . . . . . 6 ((𝑦 = (𝑧 cyclShift 𝑘) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
5 elfznn0 12471 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℕ0)
6 elfznn0 12471 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℕ0)
7 nn0addcl 11366 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℕ0)
85, 6, 7syl2anr 494 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ0)
98adantr 480 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ0)
10 elfz3nn0 12472 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...(#‘𝑧)) → (#‘𝑧) ∈ ℕ0)
1110ad2antlr 763 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ0)
12 simprl 809 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (#‘𝑧))
13 elfz2nn0 12469 . . . . . . . . . . . . . . 15 ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0 ∧ (𝑘 + 𝑚) ≤ (#‘𝑧)))
149, 11, 12, 13syl3anbrc 1265 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(#‘𝑧)))
1514adantr 480 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (𝑘 + 𝑚) ∈ (0...(#‘𝑧)))
16 cshwcsh2id.1 . . . . . . . . . . . . . . . . . 18 (𝜑𝑧 ∈ Word 𝑉)
1716adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
1817adantl 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19 elfzelz 12380 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℤ)
2019ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21 elfzelz 12380 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
2221adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 𝑚 ∈ ℤ)
2322adantr 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24 2cshw 13605 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2518, 20, 23, 24syl3anc 1366 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2625eqeq2d 2661 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2726biimpa 500 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
2815, 27jca 553 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2928exp41 637 . . . . . . . . . . 11 (𝑚 ∈ (0...(#‘𝑦)) → (𝑘 ∈ (0...(#‘𝑧)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))))
3029com23 86 . . . . . . . . . 10 (𝑚 ∈ (0...(#‘𝑦)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))))
3130com24 95 . . . . . . . . 9 (𝑚 ∈ (0...(#‘𝑦)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (𝑘 ∈ (0...(#‘𝑧)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))))
3231imp 444 . . . . . . . 8 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (𝑘 ∈ (0...(#‘𝑧)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
3332com12 32 . . . . . . 7 (𝑘 ∈ (0...(#‘𝑧)) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
3433adantl 481 . . . . . 6 ((𝑦 = (𝑧 cyclShift 𝑘) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
354, 34sylbid 230 . . . . 5 ((𝑦 = (𝑧 cyclShift 𝑘) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
3635ancoms 468 . . . 4 ((𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
3736impcom 445 . . 3 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))
38 oveq2 6698 . . . . 5 (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
3938eqeq2d 2661 . . . 4 (𝑛 = (𝑘 + 𝑚) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
4039rspcev 3340 . . 3 (((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
4137, 40syl6com 37 . 2 (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
42 elfz2 12371 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(#‘𝑧)) ↔ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (#‘𝑧))))
43 nn0z 11438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
44 zaddcl 11455 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
4544ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4645adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4746impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
48 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (#‘𝑧) ∈ ℤ)
4947, 48zsubcld 11525 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
5049ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℤ → (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5143, 50syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ0 → (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5251com12 32 . . . . . . . . . . . . . . . . . . . . 21 (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
53523adant1 1099 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (#‘𝑧))) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5542, 54sylbi 207 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
566, 55mpan9 485 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
5756adantr 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
58 elfz2nn0 12469 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...(#‘𝑧)) ↔ (𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)))
59 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
60 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑧) ∈ ℕ0 → (#‘𝑧) ∈ ℝ)
6159, 60anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
62 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
6361, 62anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
64 simplr 807 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (#‘𝑧) ∈ ℝ)
65 readdcl 10057 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6665adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6764, 66ltnled 10222 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (#‘𝑧)))
6864, 66posdifd 10652 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
6968biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7067, 69sylbird 250 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7163, 70syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7271ex 449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
73723adant3 1101 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
7458, 73sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
756, 74mpan9 485 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7675com12 32 . . . . . . . . . . . . . . . . . 18 (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7776adantr 480 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7877impcom 445 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))
79 elnnz 11425 . . . . . . . . . . . . . . . 16 (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
8057, 78, 79sylanbrc 699 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ)
8180nnnn0d 11389 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ0)
8210ad2antlr 763 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ0)
83 cshwcsh2id.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
84 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑦)) = (0...(#‘𝑧)))
8584eleq2d 2716 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑦) = (#‘𝑧) → (𝑚 ∈ (0...(#‘𝑦)) ↔ 𝑚 ∈ (0...(#‘𝑧))))
8685anbi1d 741 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ↔ (𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧)))))
87 elfz2nn0 12469 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (0...(#‘𝑧)) ↔ (𝑚 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑚 ≤ (#‘𝑧)))
8859adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → 𝑘 ∈ ℝ)
8988, 62anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
9060, 60jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝑧) ∈ ℕ0 → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
9190ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
92 le2add 10548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
9389, 91, 92syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
94 nn0readdcl 11395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9594adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9660ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (#‘𝑧) ∈ ℝ)
9795, 96, 96lesubadd2d 10664 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
9893, 97sylibrd 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
9998expcomd 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
10099ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
101100com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑘 ≤ (#‘𝑧) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
1021013impia 1280 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
103102com13 88 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → (𝑚 ≤ (#‘𝑧) → ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
104103imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ0𝑚 ≤ (#‘𝑧)) → ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
10558, 104syl5bi 232 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
1061053adant2 1100 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
10787, 106sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (0...(#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
108107imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
10986, 108syl6bi 243 . . . . . . . . . . . . . . . . . 18 ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
110109adantr 480 . . . . . . . . . . . . . . . . 17 (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
11183, 110syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
112111adantl 481 . . . . . . . . . . . . . . 15 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
113112impcom 445 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
114 elfz2nn0 12469 . . . . . . . . . . . . . 14 (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0 ∧ ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
11581, 82, 113, 114syl3anbrc 1265 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
116115adantr 480 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
11716adantl 481 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
118117adantl 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
11919ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
12022adantr 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
121118, 119, 120, 24syl3anc 1366 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
12219, 21, 44syl2anr 494 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
123 cshwsublen 13588 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
124117, 122, 123syl2anr 494 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
125121, 124eqtrd 2685 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
126125eqeq2d 2661 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
127126biimpa 500 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
128116, 127jca 553 . . . . . . . . . . 11 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
129128exp41 637 . . . . . . . . . 10 (𝑚 ∈ (0...(#‘𝑦)) → (𝑘 ∈ (0...(#‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))))))
130129com23 86 . . . . . . . . 9 (𝑚 ∈ (0...(#‘𝑦)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))))))
131130com24 95 . . . . . . . 8 (𝑚 ∈ (0...(#‘𝑦)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (𝑘 ∈ (0...(#‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))))))
132131imp 444 . . . . . . 7 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (𝑘 ∈ (0...(#‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))))
1333, 132syl6bi 243 . . . . . 6 (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑘 ∈ (0...(#‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))))))
134133com23 86 . . . . 5 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))))))
135134impcom 445 . . . 4 ((𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))))
136135impcom 445 . . 3 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))))
137 oveq2 6698 . . . . 5 (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
138137eqeq2d 2661 . . . 4 (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
139138rspcev 3340 . . 3 ((((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
140136, 139syl6com 37 . 2 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
14141, 140pm2.61ian 848 1 (𝜑 → (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  ...cfz 12364  #chash 13157  Word cword 13323   cyclShift ccsh 13580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335  df-csh 13581
This theorem is referenced by:  erclwwlktr  26979  erclwwlkntr  27035
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