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Theorem cshwcsh2id 13857
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 27258 and erclwwlkntr 27323. (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 6849 . . . . . . . . 9 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
21eqeq2d 2775 . . . . . . . 8 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
32anbi2d 622 . . . . . . 7 (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
43adantr 472 . . . . . 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 12640 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℕ0)
6 elfznn0 12640 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℕ0)
7 nn0addcl 11575 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℕ0)
85, 6, 7syl2anr 590 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ0)
98adantr 472 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ0)
10 elfz3nn0 12641 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...(♯‘𝑧)) → (♯‘𝑧) ∈ ℕ0)
1110ad2antlr 718 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ0)
12 simprl 787 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (♯‘𝑧))
13 elfz2nn0 12638 . . . . . . . . . . . . . . 15 ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0 ∧ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
149, 11, 12, 13syl3anbrc 1443 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(♯‘𝑧)))
1514adantr 472 . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 17 (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
1817adantl 473 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19 elfzelz 12549 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℤ)
2019ad2antlr 718 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21 elfzelz 12549 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
2221adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 𝑚 ∈ ℤ)
2322adantr 472 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24 2cshw 13842 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2518, 20, 23, 24syl3anc 1490 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2625eqeq2d 2775 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2726biimpa 468 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
2815, 27jca 507 . . . . . . . . . . . 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 425 . . . . . . . . . . 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 395 . . . . . . . 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 473 . . . . . 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 231 . . . . 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 450 . . . 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 396 . . 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 6850 . . . 4 (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
3938rspceeqv 3479 . . 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 12540 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(♯‘𝑧)) ↔ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (♯‘𝑧))))
42 nn0z 11647 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
43 zaddcl 11664 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
4443ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4544adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4645impcom 396 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
47 simprl 787 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (♯‘𝑧) ∈ ℤ)
4846, 47zsubcld 11734 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
4948ex 401 . . . . . . . . . . . . . . . . . . . . . . 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 1160 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5352adantr 472 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (♯‘𝑧))) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5441, 53sylbi 208 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
556, 54mpan9 502 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
5655adantr 472 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
57 elfz2nn0 12638 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...(♯‘𝑧)) ↔ (𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)))
58 nn0re 11548 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
59 nn0re 11548 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑧) ∈ ℕ0 → (♯‘𝑧) ∈ ℝ)
6058, 59anim12i 606 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
61 nn0re 11548 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
6260, 61anim12i 606 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
63 simplr 785 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (♯‘𝑧) ∈ ℝ)
64 readdcl 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6564adantlr 706 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6663, 65ltnled 10438 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
6763, 65posdifd 10868 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
6867biimpd 220 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
6966, 68sylbird 251 . . . . . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
72713adant3 1162 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
7357, 72sylbi 208 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
746, 73mpan9 502 . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7776impcom 396 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))
78 elnnz 11634 . . . . . . . . . . . . . . . 16 (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7956, 77, 78sylanbrc 578 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ)
8079nnnn0d 11598 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ0)
8110ad2antlr 718 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ0)
82 cshwcsh2id.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
83 oveq2 6850 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑦)) = (0...(♯‘𝑧)))
8483eleq2d 2830 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑦) = (♯‘𝑧) → (𝑚 ∈ (0...(♯‘𝑦)) ↔ 𝑚 ∈ (0...(♯‘𝑧))))
8584anbi1d 623 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ↔ (𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧)))))
86 elfz2nn0 12638 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (0...(♯‘𝑧)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑚 ≤ (♯‘𝑧)))
8758adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → 𝑘 ∈ ℝ)
8887, 61anim12i 606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
8959, 59jca 507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((♯‘𝑧) ∈ ℕ0 → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
9089ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
91 le2add 10764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
9288, 90, 91syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
93 nn0readdcl 11604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9493adantlr 706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9559ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (♯‘𝑧) ∈ ℝ)
9694, 95, 95lesubadd2d 10880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
9792, 96sylibrd 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
9897expcomd 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
9998ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
10457, 103syl5bi 233 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
1051043adant2 1161 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
10686, 105sylbi 208 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (0...(♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
107106imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
10885, 107syl6bi 244 . . . . . . . . . . . . . . . . . 18 ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
109108adantr 472 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 15 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
112111impcom 396 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
113 elfz2nn0 12638 . . . . . . . . . . . . . 14 (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0 ∧ ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
11480, 81, 112, 113syl3anbrc 1443 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
115114adantr 472 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
11616adantl 473 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
117116adantl 473 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
11819ad2antlr 718 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
11922adantr 472 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
120117, 118, 119, 24syl3anc 1490 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
12119, 21, 43syl2anr 590 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
122 cshwsublen 13825 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
123116, 121, 122syl2anr 590 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
124120, 123eqtrd 2799 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
125124eqeq2d 2775 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))
126125biimpa 468 . . . . . . . . . . . 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 507 . . . . . . . . . . 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 425 . . . . . . . . . 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 395 . . . . . . 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, 131syl6bi 244 . . . . . 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 396 . . . 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 396 . . 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 6850 . . . 4 (𝑛 = ((𝑘 + 𝑚) − (♯‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
137136rspceeqv 3479 . . 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 846 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3056   class class class wbr 4809  cfv 6068  (class class class)co 6842  cr 10188  0cc0 10189   + caddc 10192   < clt 10328  cle 10329  cmin 10520  cn 11274  0cn0 11538  cz 11624  ...cfz 12533  chash 13321  Word cword 13486   cyclShift ccsh 13812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-hash 13322  df-word 13487  df-concat 13542  df-substr 13617  df-pfx 13662  df-csh 13814
This theorem is referenced by:  erclwwlktr  27258  erclwwlkntr  27323
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