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Theorem cshwcsh2id 13527
 Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlkstr 26836 and erclwwlksntr 26848. (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 6622 . . . . . . . . 9 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
21eqeq2d 2631 . . . . . . . 8 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
32anbi2d 739 . . . . . . 7 (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
43adantr 481 . . . . . 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 12390 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℕ0)
6 elfznn0 12390 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℕ0)
7 nn0addcl 11288 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℕ0)
85, 6, 7syl2anr 495 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ0)
98adantr 481 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ0)
10 elfz3nn0 12391 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...(#‘𝑧)) → (#‘𝑧) ∈ ℕ0)
1110ad2antlr 762 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ0)
12 simprl 793 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (#‘𝑧))
13 elfz2nn0 12388 . . . . . . . . . . . . . . 15 ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0 ∧ (𝑘 + 𝑚) ≤ (#‘𝑧)))
149, 11, 12, 13syl3anbrc 1244 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(#‘𝑧)))
1514adantr 481 . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . 17 (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
1817adantl 482 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19 elfzelz 12300 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℤ)
2019ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21 elfzelz 12300 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
2221adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 𝑚 ∈ ℤ)
2322adantr 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24 2cshw 13512 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2518, 20, 23, 24syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2625eqeq2d 2631 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2726biimpa 501 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
2815, 27jca 554 . . . . . . . . . . . 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 445 . . . . . . . 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 482 . . . . . 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 469 . . . 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 446 . . 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 6623 . . . . 5 (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
3938eqeq2d 2631 . . . 4 (𝑛 = (𝑘 + 𝑚) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
4039rspcev 3299 . . 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 12291 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(#‘𝑧)) ↔ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (#‘𝑧))))
43 nn0z 11360 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
44 zaddcl 11377 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
4544ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4645adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4746impcom 446 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
48 simprl 793 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (#‘𝑧) ∈ ℤ)
4947, 48zsubcld 11447 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
5049ex 450 . . . . . . . . . . . . . . . . . . . . . . 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 1077 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
5453adantr 481 . . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
5756adantr 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
58 elfz2nn0 12388 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...(#‘𝑧)) ↔ (𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑘 ≤ (#‘𝑧)))
59 nn0re 11261 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
60 nn0re 11261 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑧) ∈ ℕ0 → (#‘𝑧) ∈ ℝ)
6159, 60anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
62 nn0re 11261 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
6361, 62anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
64 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (#‘𝑧) ∈ ℝ)
65 readdcl 9979 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6665adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6764, 66ltnled 10144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (#‘𝑧)))
6864, 66posdifd 10574 . . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
73723adant3 1079 . . . . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
7877impcom 446 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))
79 elnnz 11347 . . . . . . . . . . . . . . . 16 (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
8057, 78, 79sylanbrc 697 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ)
8180nnnn0d 11311 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ0)
8210ad2antlr 762 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ0)
83 cshwcsh2id.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
84 oveq2 6623 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑦)) = (0...(#‘𝑧)))
8584eleq2d 2684 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑦) = (#‘𝑧) → (𝑚 ∈ (0...(#‘𝑦)) ↔ 𝑚 ∈ (0...(#‘𝑧))))
8685anbi1d 740 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ↔ (𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧)))))
87 elfz2nn0 12388 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (0...(#‘𝑧)) ↔ (𝑚 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0𝑚 ≤ (#‘𝑧)))
8859adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) → 𝑘 ∈ ℝ)
8988, 62anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
9060, 60jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((#‘𝑧) ∈ ℕ0 → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
9190ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
92 le2add 10470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
9389, 91, 92syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
94 nn0readdcl 11317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9594adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9660ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (#‘𝑧) ∈ ℝ)
9795, 96, 96lesubadd2d 10586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑘 ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
10099ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1258 . . . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . . 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 1078 . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
113112impcom 446 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
114 elfz2nn0 12388 . . . . . . . . . . . . . 14 (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ0 ∧ (#‘𝑧) ∈ ℕ0 ∧ ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
11581, 82, 113, 114syl3anbrc 1244 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
116115adantr 481 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
11716adantl 482 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
118117adantl 482 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
11919ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
12022adantr 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
121118, 119, 120, 24syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
12219, 21, 44syl2anr 495 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
123 cshwsublen 13495 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
124117, 122, 123syl2anr 495 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
125121, 124eqtrd 2655 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
126125eqeq2d 2631 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
127126biimpa 501 . . . . . . . . . . . 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 554 . . . . . . . . . . 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 445 . . . . . . 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 446 . . . 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 446 . . 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 6623 . . . . 5 (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
138137eqeq2d 2631 . . . 4 (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
139138rspcev 3299 . . 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 830 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∃wrex 2909   class class class wbr 4623  ‘cfv 5857  (class class class)co 6615  ℝcr 9895  0cc0 9896   + caddc 9899   < clt 10034   ≤ cle 10035   − cmin 10226  ℕcn 10980  ℕ0cn0 11252  ℤcz 11337  ...cfz 12284  #chash 13073  Word cword 13246   cyclShift ccsh 13487 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-hash 13074  df-word 13254  df-concat 13256  df-substr 13258  df-csh 13488 This theorem is referenced by:  erclwwlkstr  26836  erclwwlksntr  26848
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