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Theorem cshwcsh2id 13367
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 26105 and erclwwlkntr 26117. (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 6530 . . . . . . . . 9 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
21eqeq2d 2615 . . . . . . . 8 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
32anbi2d 735 . . . . . . 7 (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
43adantr 479 . . . . . 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 12253 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℕ0)
6 elfznn0 12253 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℕ0)
7 nn0addcl 11171 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℕ0)
85, 6, 7syl2anr 493 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ0)
98adantr 479 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ0)
10 elfz3nn0 12254 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...(#‘𝑧)) → (#‘𝑧) ∈ ℕ0)
1110ad2antlr 758 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ0)
12 simprl 789 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (#‘𝑧))
13 elfz2nn0 12251 . . . . . . . . . . . . . . 15 ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0 ∧ (𝑘 + 𝑚) ≤ (#‘𝑧)))
149, 11, 12, 13syl3anbrc 1238 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(#‘𝑧)))
1514adantr 479 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
1817adantl 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19 elfzelz 12164 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℤ)
2019ad2antlr 758 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21 elfzelz 12164 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
2221adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 𝑚 ∈ ℤ)
2322adantr 479 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24 2cshw 13352 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2518, 20, 23, 24syl3anc 1317 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2625eqeq2d 2615 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2726biimpa 499 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
2815, 27jca 552 . . . . . . . . . . . 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 635 . . . . . . . . . . 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 83 . . . . . . . . . 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 92 . . . . . . . . 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 443 . . . . . . . 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 480 . . . . . 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 228 . . . . 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 467 . . . 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 444 . . 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 6531 . . . . 5 (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
3938eqeq2d 2615 . . . 4 (𝑛 = (𝑘 + 𝑚) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
4039rspcev 3277 . . 3 (((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
4137, 40syl6com 36 . 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 12155 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(#‘𝑧)) ↔ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (#‘𝑧))))
43 nn0z 11229 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
44 zaddcl 11246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
4544ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4645adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4746impcom 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
48 simprl 789 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (#‘𝑧) ∈ ℤ)
4947, 48zsubcld 11315 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
5049ex 448 . . . . . . . . . . . . . . . . . . . . . . 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 1071 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5453adantr 479 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (#‘𝑧))) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5542, 54sylbi 205 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
566, 55mpan9 484 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
5756adantr 479 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
58 elfz2nn0 12251 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...(#‘𝑧)) ↔ (𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)))
59 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
60 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑧) ∈ ℕ0 → (#‘𝑧) ∈ ℝ)
6159, 60anim12i 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
62 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
6361, 62anim12i 587 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
64 simplr 787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (#‘𝑧) ∈ ℝ)
65 readdcl 9871 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6665adantlr 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6764, 66ltnled 10031 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (#‘𝑧)))
6864, 66posdifd 10459 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
6968biimpd 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7067, 69sylbird 248 . . . . . . . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
73723adant3 1073 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
7458, 73sylbi 205 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
756, 74mpan9 484 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7877impcom 444 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))
79 elnnz 11216 . . . . . . . . . . . . . . . 16 (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
8057, 78, 79sylanbrc 694 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ)
8180nnnn0d 11194 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ0)
8210ad2antlr 758 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ0)
83 cshwcsh2id.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
84 oveq2 6531 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑦)) = (0...(#‘𝑧)))
8584eleq2d 2668 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑦) = (#‘𝑧) → (𝑚 ∈ (0...(#‘𝑦)) ↔ 𝑚 ∈ (0...(#‘𝑧))))
8685anbi1d 736 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ↔ (𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧)))))
87 elfz2nn0 12251 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (0...(#‘𝑧)) ↔ (𝑚 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑚 ≤ (#‘𝑧)))
8859adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → 𝑘 ∈ ℝ)
8988, 62anim12i 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
9060, 60jca 552 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝑧) ∈ ℕ0 → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
9190ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
92 le2add 10355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
9389, 91, 92syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
94 nn0readdcl 11200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9594adantlr 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9660ad2antlr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (#‘𝑧) ∈ ℝ)
9795, 96, 96lesubadd2d 10471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
9893, 97sylibrd 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
9998expcomd 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
10099ex 448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
101100com24 92 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑘 ≤ (#‘𝑧) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
1021013impia 1252 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
103102com13 85 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → (𝑚 ≤ (#‘𝑧) → ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
104103imp 443 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ0𝑚 ≤ (#‘𝑧)) → ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
10558, 104syl5bi 230 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
1061053adant2 1072 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
10787, 106sylbi 205 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (0...(#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
108107imp 443 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
10986, 108syl6bi 241 . . . . . . . . . . . . . . . . . 18 ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
110109adantr 479 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
113112impcom 444 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
114 elfz2nn0 12251 . . . . . . . . . . . . . 14 (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0 ∧ ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
11581, 82, 113, 114syl3anbrc 1238 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
116115adantr 479 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
11716adantl 480 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
118117adantl 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
11919ad2antlr 758 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
12022adantr 479 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
121118, 119, 120, 24syl3anc 1317 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
12219, 21, 44syl2anr 493 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
123 cshwsublen 13335 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
124117, 122, 123syl2anr 493 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
125121, 124eqtrd 2639 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
126125eqeq2d 2615 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
127126biimpa 499 . . . . . . . . . . . 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 552 . . . . . . . . . . 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 635 . . . . . . . . . 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 83 . . . . . . . . 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 92 . . . . . . . 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 443 . . . . . . 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 241 . . . . . 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 83 . . . . 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 444 . . . 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 444 . . 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 6531 . . . . 5 (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
138137eqeq2d 2615 . . . 4 (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
139138rspcev 3277 . . 3 ((((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
140136, 139syl6com 36 . 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 826 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2892   class class class wbr 4573  cfv 5786  (class class class)co 6523  cr 9787  0cc0 9788   + caddc 9791   < clt 9926  cle 9927  cmin 10113  cn 10863  0cn0 11135  cz 11206  ...cfz 12148  #chash 12930  Word cword 13088   cyclShift ccsh 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-csh 13328
This theorem is referenced by:  erclwwlktr  26105  erclwwlkntr  26117  erclwwlkstr  41241  erclwwlksntr  41253
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