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Theorem cshwcsh2id 14178
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 27727 and erclwwlkntr 27777. (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 7152 . . . . . . . . 9 (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
21eqeq2d 2829 . . . . . . . 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 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 12988 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℕ0)
6 elfznn0 12988 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℕ0)
7 nn0addcl 11920 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℕ0)
85, 6, 7syl2anr 596 . . . . . . . . . . . . . . . 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 12989 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...(♯‘𝑧)) → (♯‘𝑧) ∈ ℕ0)
1110ad2antlr 723 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ0)
12 simprl 767 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (♯‘𝑧))
13 elfz2nn0 12986 . . . . . . . . . . . . . . 15 ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0 ∧ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
149, 11, 12, 13syl3anbrc 1335 . . . . . . . . . . . . . 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 12896 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℤ)
2019ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21 elfzelz 12896 . . . . . . . . . . . . . . . . . 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 14163 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ Word 𝑉𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2518, 20, 23, 24syl3anc 1363 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
2625eqeq2d 2829 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
2726biimpa 477 . . . . . . . . . . . . 13 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
2815, 27jca 512 . . . . . . . . . . . 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 435 . . . . . . . . . . 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 407 . . . . . . . 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 241 . . . . 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 459 . . . 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 408 . . 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 7153 . . . 4 (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
3938rspceeqv 3635 . . 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 12887 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(♯‘𝑧)) ↔ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (♯‘𝑧))))
42 nn0z 11993 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
43 zaddcl 12010 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
4443ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4544adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
4645impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
47 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (♯‘𝑧) ∈ ℤ)
4846, 47zsubcld 12080 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
4948ex 413 . . . . . . . . . . . . . . . . . . . . . . 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 1122 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5352adantr 481 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘 ≤ (♯‘𝑧))) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
5441, 53sylbi 218 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ0 → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
556, 54mpan9 507 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
5655adantr 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
57 elfz2nn0 12986 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...(♯‘𝑧)) ↔ (𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)))
58 nn0re 11894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
59 nn0re 11894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑧) ∈ ℕ0 → (♯‘𝑧) ∈ ℝ)
6058, 59anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
61 nn0re 11894 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
6260, 61anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
63 simplr 765 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (♯‘𝑧) ∈ ℝ)
64 readdcl 10608 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6564adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
6663, 65ltnled 10775 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
6763, 65posdifd 11215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
6867biimpd 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
6966, 68sylbird 261 . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
72713adant3 1124 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
7357, 72sylbi 218 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ0 → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
746, 73mpan9 507 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7776impcom 408 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))
78 elnnz 11979 . . . . . . . . . . . . . . . 16 (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
7956, 77, 78sylanbrc 583 . . . . . . . . . . . . . . 15 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ)
8079nnnn0d 11943 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ0)
8110ad2antlr 723 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ0)
82 cshwcsh2id.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
83 oveq2 7153 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑦)) = (0...(♯‘𝑧)))
8483eleq2d 2895 . . . . . . . . . . . . . . . . . . . 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 12986 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (0...(♯‘𝑧)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑚 ≤ (♯‘𝑧)))
8758adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) → 𝑘 ∈ ℝ)
8887, 61anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
8959, 59jca 512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((♯‘𝑧) ∈ ℕ0 → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
9089ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
91 le2add 11110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
9288, 90, 91syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
93 nn0readdcl 11949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ ℕ0𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9493adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑘 + 𝑚) ∈ ℝ)
9559ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (♯‘𝑧) ∈ ℝ)
9694, 95, 95lesubadd2d 11227 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
9792, 96sylibrd 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
9897expcomd 417 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0) ∧ 𝑚 ∈ ℕ0) → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
9998ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → ((𝑘 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
10457, 103syl5bi 243 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
1051043adant2 1123 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
10686, 105sylbi 218 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (0...(♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
107106imp 407 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
10885, 107syl6bi 254 . . . . . . . . . . . . . . . . . 18 ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
109108adantr 481 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
112111impcom 408 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
113 elfz2nn0 12986 . . . . . . . . . . . . . 14 (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ0 ∧ (♯‘𝑧) ∈ ℕ0 ∧ ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
11480, 81, 112, 113syl3anbrc 1335 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
115114adantr 481 . . . . . . . . . . . 12 ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
11616adantl 482 . . . . . . . . . . . . . . . . 17 ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
117116adantl 482 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
11819ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
11922adantr 481 . . . . . . . . . . . . . . . 16 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
120117, 118, 119, 24syl3anc 1363 . . . . . . . . . . . . . . 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 14146 . . . . . . . . . . . . . . . 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 2853 . . . . . . . . . . . . . 14 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
125124eqeq2d 2829 . . . . . . . . . . . . 13 (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))
126125biimpa 477 . . . . . . . . . . . 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 512 . . . . . . . . . . 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 435 . . . . . . . . . 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 407 . . . . . . 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 254 . . . . . 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 408 . . . 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 408 . . 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 7153 . . . 4 (𝑛 = ((𝑘 + 𝑚) − (♯‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
137136rspceeqv 3635 . . 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 808 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525   + caddc 10528   < clt 10663  cle 10664  cmin 10858  cn 11626  0cn0 11885  cz 11969  ...cfz 12880  chash 13678  Word cword 13849   cyclShift ccsh 14138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-hash 13679  df-word 13850  df-concat 13911  df-substr 13991  df-pfx 14021  df-csh 14139
This theorem is referenced by:  erclwwlktr  27727  erclwwlkntr  27777
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