Step |
Hyp |
Ref |
Expression |
1 |
|
cshwcsh2id.1 |
⊢ ( 𝜑 → 𝑧 ∈ Word 𝑉 ) |
2 |
|
cshwcsh2id.2 |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) ) |
3 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ( 𝑦 cyclShift 𝑚 ) = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) |
4 |
3
|
eqeq2d |
⊢ ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ( 𝑥 = ( 𝑦 cyclShift 𝑚 ) ↔ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) ) |
5 |
4
|
anbi2d |
⊢ ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ↔ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑦 = ( 𝑧 cyclShift 𝑘 ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ↔ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) ) ) |
7 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → 𝑘 ∈ ℕ0 ) |
8 |
|
elfznn0 |
⊢ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) → 𝑚 ∈ ℕ0 ) |
9 |
|
nn0addcl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ) → ( 𝑘 + 𝑚 ) ∈ ℕ0 ) |
10 |
7 8 9
|
syl2anr |
⊢ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( 𝑘 + 𝑚 ) ∈ ℕ0 ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( 𝑘 + 𝑚 ) ∈ ℕ0 ) |
12 |
|
elfz3nn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) |
13 |
12
|
ad2antlr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) |
14 |
|
simprl |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ) |
15 |
|
elfz2nn0 |
⊢ ( ( 𝑘 + 𝑚 ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ↔ ( ( 𝑘 + 𝑚 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
16 |
11 13 14 15
|
syl3anbrc |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( 𝑘 + 𝑚 ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) ∧ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) → ( 𝑘 + 𝑚 ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) |
18 |
1
|
adantl |
⊢ ( ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) → 𝑧 ∈ Word 𝑉 ) |
19 |
18
|
adantl |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 𝑧 ∈ Word 𝑉 ) |
20 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → 𝑘 ∈ ℤ ) |
21 |
20
|
ad2antlr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 𝑘 ∈ ℤ ) |
22 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) → 𝑚 ∈ ℤ ) |
23 |
22
|
adantr |
⊢ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → 𝑚 ∈ ℤ ) |
24 |
23
|
adantr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 𝑚 ∈ ℤ ) |
25 |
|
2cshw |
⊢ ( ( 𝑧 ∈ Word 𝑉 ∧ 𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ ) → ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) |
26 |
19 21 24 25
|
syl3anc |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) |
27 |
26
|
eqeq2d |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ↔ 𝑥 = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) ) |
28 |
27
|
biimpa |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) ∧ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) → 𝑥 = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) |
29 |
17 28
|
jca |
⊢ ( ( ( ( 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 ) ) ) ) |
30 |
29
|
exp41 |
⊢ ( 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 ) ) ) ) ) ) ) |
31 |
30
|
com23 |
⊢ ( 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 ) ) ) ) ) ) ) |
32 |
31
|
com24 |
⊢ ( 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 ) ) ) ) ) ) ) |
33 |
32
|
imp |
⊢ ( ( 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 ) ) ) ) ) ) |
34 |
33
|
com12 |
⊢ ( 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 ) ) ) ) ) ) |
35 |
34
|
adantl |
⊢ ( ( 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 ) ) ) ) ) ) |
36 |
6 35
|
sylbid |
⊢ ( ( 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 ) ) ) ) ) ) |
37 |
36
|
ancoms |
⊢ ( ( 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 ) ) ) ) ) ) |
38 |
37
|
impcom |
⊢ ( ( ( 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 ) ) ) ) ) |
39 |
|
oveq2 |
⊢ ( 𝑛 = ( 𝑘 + 𝑚 ) → ( 𝑧 cyclShift 𝑛 ) = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) |
40 |
39
|
rspceeqv |
⊢ ( ( ( 𝑘 + 𝑚 ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑥 = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) |
41 |
38 40
|
syl6com |
⊢ ( ( ( 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 |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ↔ ( ( 0 ∈ ℤ ∧ ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 0 ≤ 𝑘 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) ) ) |
43 |
|
nn0z |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ ) |
44 |
|
zaddcl |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ ) → ( 𝑘 + 𝑚 ) ∈ ℤ ) |
45 |
44
|
ex |
⊢ ( 𝑘 ∈ ℤ → ( 𝑚 ∈ ℤ → ( 𝑘 + 𝑚 ) ∈ ℤ ) ) |
46 |
45
|
adantl |
⊢ ( ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑚 ∈ ℤ → ( 𝑘 + 𝑚 ) ∈ ℤ ) ) |
47 |
46
|
impcom |
⊢ ( ( 𝑚 ∈ ℤ ∧ ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) ) → ( 𝑘 + 𝑚 ) ∈ ℤ ) |
48 |
|
simprl |
⊢ ( ( 𝑚 ∈ ℤ ∧ ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) ) → ( ♯ ‘ 𝑧 ) ∈ ℤ ) |
49 |
47 48
|
zsubcld |
⊢ ( ( 𝑚 ∈ ℤ ∧ ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) |
50 |
49
|
ex |
⊢ ( 𝑚 ∈ ℤ → ( ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) ) |
51 |
43 50
|
syl |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) ) |
52 |
51
|
com12 |
⊢ ( ( ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑚 ∈ ℕ0 → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) ) |
53 |
52
|
3adant1 |
⊢ ( ( 0 ∈ ℤ ∧ ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) → ( 𝑚 ∈ ℕ0 → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) ) |
54 |
53
|
adantr |
⊢ ( ( ( 0 ∈ ℤ ∧ ( ♯ ‘ 𝑧 ) ∈ ℤ ∧ 𝑘 ∈ ℤ ) ∧ ( 0 ≤ 𝑘 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) ) → ( 𝑚 ∈ ℕ0 → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) ) |
55 |
42 54
|
sylbi |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( 𝑚 ∈ ℕ0 → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) ) |
56 |
8 55
|
mpan9 |
⊢ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) |
57 |
56
|
adantr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ) |
58 |
|
elfz2nn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ↔ ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) ) |
59 |
|
nn0re |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ ) |
60 |
|
nn0re |
⊢ ( ( ♯ ‘ 𝑧 ) ∈ ℕ0 → ( ♯ ‘ 𝑧 ) ∈ ℝ ) |
61 |
59 60
|
anim12i |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) → ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ) |
62 |
|
nn0re |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℝ ) |
63 |
61 62
|
anim12i |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) ) |
64 |
|
simplr |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ♯ ‘ 𝑧 ) ∈ ℝ ) |
65 |
|
readdcl |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ) → ( 𝑘 + 𝑚 ) ∈ ℝ ) |
66 |
65
|
adantlr |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( 𝑘 + 𝑚 ) ∈ ℝ ) |
67 |
64 66
|
ltnled |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ( ♯ ‘ 𝑧 ) < ( 𝑘 + 𝑚 ) ↔ ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
68 |
64 66
|
posdifd |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ( ♯ ‘ 𝑧 ) < ( 𝑘 + 𝑚 ) ↔ 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
69 |
68
|
biimpd |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ( ♯ ‘ 𝑧 ) < ( 𝑘 + 𝑚 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
70 |
67 69
|
sylbird |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
71 |
63 70
|
syl |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
72 |
71
|
ex |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) → ( 𝑚 ∈ ℕ0 → ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) ) |
73 |
72
|
3adant3 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) → ( 𝑚 ∈ ℕ0 → ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) ) |
74 |
58 73
|
sylbi |
⊢ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( 𝑚 ∈ ℕ0 → ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) ) |
75 |
8 74
|
mpan9 |
⊢ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
76 |
75
|
com12 |
⊢ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
77 |
76
|
adantr |
⊢ ( ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
78 |
77
|
impcom |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) |
79 |
|
elnnz |
⊢ ( ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℕ ↔ ( ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℤ ∧ 0 < ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
80 |
57 78 79
|
sylanbrc |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℕ ) |
81 |
80
|
nnnn0d |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℕ0 ) |
82 |
12
|
ad2antlr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) |
83 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( 0 ... ( ♯ ‘ 𝑦 ) ) = ( 0 ... ( ♯ ‘ 𝑧 ) ) ) |
84 |
83
|
eleq2d |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ↔ 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ) |
85 |
84
|
anbi1d |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ↔ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ) ) |
86 |
|
elfz2nn0 |
⊢ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ↔ ( 𝑚 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) ) |
87 |
59
|
adantr |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) → 𝑘 ∈ ℝ ) |
88 |
87 62
|
anim12i |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( 𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ) ) |
89 |
60 60
|
jca |
⊢ ( ( ♯ ‘ 𝑧 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑧 ) ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ) |
90 |
89
|
ad2antlr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ( ♯ ‘ 𝑧 ) ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ) |
91 |
|
le2add |
⊢ ( ( ( 𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ) ∧ ( ( ♯ ‘ 𝑧 ) ∈ ℝ ∧ ( ♯ ‘ 𝑧 ) ∈ ℝ ) ) → ( ( 𝑘 ≤ ( ♯ ‘ 𝑧 ) ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) → ( 𝑘 + 𝑚 ) ≤ ( ( ♯ ‘ 𝑧 ) + ( ♯ ‘ 𝑧 ) ) ) ) |
92 |
88 90 91
|
syl2anc |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝑘 ≤ ( ♯ ‘ 𝑧 ) ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) → ( 𝑘 + 𝑚 ) ≤ ( ( ♯ ‘ 𝑧 ) + ( ♯ ‘ 𝑧 ) ) ) ) |
93 |
|
nn0readdcl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ) → ( 𝑘 + 𝑚 ) ∈ ℝ ) |
94 |
93
|
adantlr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( 𝑘 + 𝑚 ) ∈ ℝ ) |
95 |
60
|
ad2antlr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ♯ ‘ 𝑧 ) ∈ ℝ ) |
96 |
94 95 95
|
lesubadd2d |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ↔ ( 𝑘 + 𝑚 ) ≤ ( ( ♯ ‘ 𝑧 ) + ( ♯ ‘ 𝑧 ) ) ) ) |
97 |
92 96
|
sylibrd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( ( 𝑘 ≤ ( ♯ ‘ 𝑧 ) ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
98 |
97
|
expcomd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) ∧ 𝑚 ∈ ℕ0 ) → ( 𝑚 ≤ ( ♯ ‘ 𝑧 ) → ( 𝑘 ≤ ( ♯ ‘ 𝑧 ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) ) |
99 |
98
|
ex |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) → ( 𝑚 ∈ ℕ0 → ( 𝑚 ≤ ( ♯ ‘ 𝑧 ) → ( 𝑘 ≤ ( ♯ ‘ 𝑧 ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) ) ) |
100 |
99
|
com24 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ) → ( 𝑘 ≤ ( ♯ ‘ 𝑧 ) → ( 𝑚 ≤ ( ♯ ‘ 𝑧 ) → ( 𝑚 ∈ ℕ0 → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) ) ) |
101 |
100
|
3impia |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) → ( 𝑚 ≤ ( ♯ ‘ 𝑧 ) → ( 𝑚 ∈ ℕ0 → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) ) |
102 |
101
|
com13 |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 ≤ ( ♯ ‘ 𝑧 ) → ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) ) |
103 |
102
|
imp |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑘 ≤ ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
104 |
58 103
|
syl5bi |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) → ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
105 |
104
|
3adant2 |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ 𝑚 ≤ ( ♯ ‘ 𝑧 ) ) → ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
106 |
86 105
|
sylbi |
⊢ ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
107 |
106
|
imp |
⊢ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) |
108 |
85 107
|
syl6bi |
⊢ ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
109 |
108
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑦 ) = ( ♯ ‘ 𝑧 ) ∧ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
110 |
2 109
|
syl |
⊢ ( 𝜑 → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
111 |
110
|
adantl |
⊢ ( ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) → ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
112 |
111
|
impcom |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) |
113 |
|
elfz2nn0 |
⊢ ( ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ↔ ( ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑧 ) ∈ ℕ0 ∧ ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ≤ ( ♯ ‘ 𝑧 ) ) ) |
114 |
81 82 112 113
|
syl3anbrc |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) |
115 |
114
|
adantr |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) ∧ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) → ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) |
116 |
1
|
adantl |
⊢ ( ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) → 𝑧 ∈ Word 𝑉 ) |
117 |
116
|
adantl |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 𝑧 ∈ Word 𝑉 ) |
118 |
20
|
ad2antlr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 𝑘 ∈ ℤ ) |
119 |
23
|
adantr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → 𝑚 ∈ ℤ ) |
120 |
117 118 119 25
|
syl3anc |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) = ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) ) |
121 |
20 22 44
|
syl2anr |
⊢ ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) → ( 𝑘 + 𝑚 ) ∈ ℤ ) |
122 |
|
cshwsublen |
⊢ ( ( 𝑧 ∈ Word 𝑉 ∧ ( 𝑘 + 𝑚 ) ∈ ℤ ) → ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
123 |
116 121 122
|
syl2anr |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( 𝑧 cyclShift ( 𝑘 + 𝑚 ) ) = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
124 |
120 123
|
eqtrd |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
125 |
124
|
eqeq2d |
⊢ ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) → ( 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ↔ 𝑥 = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) ) |
126 |
125
|
biimpa |
⊢ ( ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ) ∧ ( ¬ ( 𝑘 + 𝑚 ) ≤ ( ♯ ‘ 𝑧 ) ∧ 𝜑 ) ) ∧ 𝑥 = ( ( 𝑧 cyclShift 𝑘 ) cyclShift 𝑚 ) ) → 𝑥 = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
127 |
115 126
|
jca |
⊢ ( ( ( ( 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 ) ) ) ) ) |
128 |
127
|
exp41 |
⊢ ( 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 ) ) ) ) ) ) ) ) |
129 |
128
|
com23 |
⊢ ( 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 ) ) ) ) ) ) ) ) |
130 |
129
|
com24 |
⊢ ( 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 ) ) ) ) ) ) ) ) |
131 |
130
|
imp |
⊢ ( ( 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 ) ) ) ) ) ) ) |
132 |
5 131
|
syl6bi |
⊢ ( 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 ) ) ) ) ) ) ) ) |
133 |
132
|
com23 |
⊢ ( 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 ) ) ) ) ) ) ) ) |
134 |
133
|
impcom |
⊢ ( ( 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 ) ) ) ) ) ) ) |
135 |
134
|
impcom |
⊢ ( ( ( 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 |
⊢ ( 𝑛 = ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) → ( 𝑧 cyclShift 𝑛 ) = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) |
137 |
136
|
rspceeqv |
⊢ ( ( ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑥 = ( 𝑧 cyclShift ( ( 𝑘 + 𝑚 ) − ( ♯ ‘ 𝑧 ) ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) |
138 |
135 137
|
syl6com |
⊢ ( ( ¬ ( 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 ) ) ) |
139 |
41 138
|
pm2.61ian |
⊢ ( 𝜑 → ( ( ( 𝑚 ∈ ( 0 ... ( ♯ ‘ 𝑦 ) ) ∧ 𝑥 = ( 𝑦 cyclShift 𝑚 ) ) ∧ ( 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) ∧ 𝑦 = ( 𝑧 cyclShift 𝑘 ) ) ) → ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑧 ) ) 𝑥 = ( 𝑧 cyclShift 𝑛 ) ) ) |