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Theorem cshwcsh2id 14803
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 29819 and erclwwlkntr 29868. (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 7421 . . . . . . . . 9 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
21eqeq2d 2738 . . . . . . . 8 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
32anbi2d 628 . . . . . . 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 13618 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℕ0)
6 elfznn0 13618 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℕ0)
7 nn0addcl 12529 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℕ0)
85, 6, 7syl2anr 596 . . . . . . . . . . . . . . . 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 13619 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...(♯‘𝑧)) → (♯‘𝑧) ∈ ℕ0)
1110ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ0)
12 simprl 770 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (♯‘𝑧))
13 elfz2nn0 13616 . . . . . . . . . . . . . . 15 ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0 ∧ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
149, 11, 12, 13syl3anbrc 1341 . . . . . . . . . . . . . 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 13525 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℤ)
2019ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21 elfzelz 13525 . . . . . . . . . . . . . . . . . 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 14787 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2518, 20, 23, 24syl3anc 1369 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2625eqeq2d 2738 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2726biimpa 476 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
2815, 27jca 511 . . . . . . . . . . . 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 434 . . . . . . . . . . 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 406 . . . . . . . 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 239 . . . . 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 458 . . . 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 407 . . 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 7422 . . . 4 (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
3938rspceeqv 3629 . . 3 (((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
4037, 39syl6com 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)))
41 elfz2 13515 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(♯‘𝑧)) ↔ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (♯‘𝑧))))
42 nn0z 12605 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
43 zaddcl 12624 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
4443ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4544adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4645impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
47 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (♯‘𝑧) ∈ ℤ)
4846, 47zsubcld 12693 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
4948ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℤ → (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5042, 49syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ0 → (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5150com12 32 . . . . . . . . . . . . . . . . . . . . 21 (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
52513adant1 1128 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5352adantr 480 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (♯‘𝑧))) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5441, 53sylbi 216 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
556, 54mpan9 506 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
5655adantr 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
57 elfz2nn0 13616 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...(♯‘𝑧)) ↔ (𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)))
58 nn0re 12503 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
59 nn0re 12503 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑧) ∈ ℕ0 → (♯‘𝑧) ∈ ℝ)
6058, 59anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
61 nn0re 12503 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
6260, 61anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
63 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (♯‘𝑧) ∈ ℝ)
64 readdcl 11213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6564adantlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6663, 65ltnled 11383 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
6763, 65posdifd 11823 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
6867biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
6966, 68sylbird 260 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7062, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7170ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
72713adant3 1130 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
7357, 72sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
746, 73mpan9 506 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7574com12 32 . . . . . . . . . . . . . . . . . 18 (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7675adantr 480 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7776impcom 407 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))
78 elnnz 12590 . . . . . . . . . . . . . . . 16 (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7956, 77, 78sylanbrc 582 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ)
8079nnnn0d 12554 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ0)
8110ad2antlr 726 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ0)
82 cshwcsh2id.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
83 oveq2 7422 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑦)) = (0...(♯‘𝑧)))
8483eleq2d 2814 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑦) = (♯‘𝑧) → (𝑚 ∈ (0...(♯‘𝑦)) ↔ 𝑚 ∈ (0...(♯‘𝑧))))
8584anbi1d 629 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ↔ (𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧)))))
86 elfz2nn0 13616 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (0...(♯‘𝑧)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑚 ≤ (♯‘𝑧)))
8758adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → 𝑘 ∈ ℝ)
8887, 61anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
8959, 59jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((♯‘𝑧) ∈ ℕ0 → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
9089ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
91 le2add 11718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
9288, 90, 91syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
93 nn0readdcl 12560 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9493adantlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9559ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (♯‘𝑧) ∈ ℝ)
9694, 95, 95lesubadd2d 11835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
9792, 96sylibrd 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
9897expcomd 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
9998ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))))
10099com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑘 ≤ (♯‘𝑧) → (𝑚 ≤ (♯‘𝑧) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))))
1011003impia 1115 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → (𝑚 ≤ (♯‘𝑧) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
102101com13 88 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → (𝑚 ≤ (♯‘𝑧) → ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
103102imp 406 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
10457, 103biimtrid 241 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
1051043adant2 1129 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
10686, 105sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (0...(♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
107106imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
10885, 107biimtrdi 252 . . . . . . . . . . . . . . . . . 18 ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
109108adantr 480 . . . . . . . . . . . . . . . . 17 (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
11082, 109syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
111110adantl 481 . . . . . . . . . . . . . . 15 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
112111impcom 407 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
113 elfz2nn0 13616 . . . . . . . . . . . . . 14 (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0 ∧ ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
11480, 81, 112, 113syl3anbrc 1341 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
115114adantr 480 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
11616adantl 481 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
117116adantl 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
11819ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
11922adantr 480 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
120117, 118, 119, 24syl3anc 1369 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
12119, 21, 43syl2anr 596 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
122 cshwsublen 14770 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
123116, 121, 122syl2anr 596 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
124120, 123eqtrd 2767 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
125124eqeq2d 2738 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))
126125biimpa 476 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
127115, 126jca 511 . . . . . . . . . . 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)))))
128127exp41 434 . . . . . . . . . 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))))))))
129128com23 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))))))))
130129com24 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))))))))
131130imp 406 . . . . . . 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)))))))
1323, 131biimtrdi 252 . . . . . 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))))))))
133132com23 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))))))))
134133impcom 407 . . . 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)))))))
135134impcom 407 . . 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))))))
136 oveq2 7422 . . . 4 (𝑛 = ((𝑘 + 𝑚) − (♯‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
137136rspceeqv 3629 . . 3 ((((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
138135, 137syl6com 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)))
13940, 138pm2.61ian 811 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wrex 3065   class class class wbr 5142  cfv 6542  (class class class)co 7414  cr 11129  0cc0 11130   + caddc 11133   < clt 11270  cle 11271  cmin 11466  cn 12234  0cn0 12494  cz 12580  ...cfz 13508  chash 14313  Word cword 14488   cyclShift ccsh 14762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-n0 12495  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-hash 14314  df-word 14489  df-concat 14545  df-substr 14615  df-pfx 14645  df-csh 14763
This theorem is referenced by:  erclwwlktr  29819  erclwwlkntr  29868
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