| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsumwrd2dccatlem.u |
⊢ 𝑈 = ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) |
| 2 |
|
gsumwrd2dccatlem.f |
⊢ 𝐹 = ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ) |
| 3 |
|
gsumwrd2dccatlem.g |
⊢ 𝐺 = ( 𝑏 ∈ 𝑈 ↦ 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ) |
| 4 |
|
gsumwrd2dccatlem.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
| 5 |
|
sneq |
⊢ ( 𝑤 = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) → { 𝑤 } = { ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) } ) |
| 6 |
|
fveq2 |
⊢ ( 𝑤 = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) |
| 7 |
6
|
oveq2d |
⊢ ( 𝑤 = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) → ( 0 ... ( ♯ ‘ 𝑤 ) ) = ( 0 ... ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) ) |
| 8 |
5 7
|
xpeq12d |
⊢ ( 𝑤 = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) → ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) = ( { ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) } × ( 0 ... ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) ) ) |
| 9 |
8
|
eleq2d |
⊢ ( 𝑤 = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) → ( 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ∈ ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ↔ 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ∈ ( { ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) } × ( 0 ... ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) ) ) ) |
| 10 |
|
xp1st |
⊢ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
| 11 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
| 12 |
|
xp2nd |
⊢ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
| 13 |
12
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
| 14 |
|
ccatcl |
⊢ ( ( ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ∧ ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) → ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∈ Word 𝐴 ) |
| 15 |
11 13 14
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∈ Word 𝐴 ) |
| 16 |
|
ovex |
⊢ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∈ V |
| 17 |
16
|
snid |
⊢ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∈ { ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) } |
| 18 |
17
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∈ { ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) } ) |
| 19 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 0 ∈ ℤ ) |
| 20 |
|
lencl |
⊢ ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∈ Word 𝐴 → ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∈ ℕ0 ) |
| 21 |
15 20
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∈ ℕ0 ) |
| 22 |
21
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∈ ℤ ) |
| 23 |
|
lencl |
⊢ ( ( 1st ‘ 𝑎 ) ∈ Word 𝐴 → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ∈ ℕ0 ) |
| 24 |
11 23
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ∈ ℕ0 ) |
| 25 |
24
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ∈ ℤ ) |
| 26 |
24
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 0 ≤ ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) |
| 27 |
|
lencl |
⊢ ( ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 → ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ∈ ℕ0 ) |
| 28 |
13 27
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ∈ ℕ0 ) |
| 29 |
28
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 0 ≤ ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) |
| 30 |
24
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ∈ ℝ ) |
| 31 |
28
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ∈ ℝ ) |
| 32 |
30 31
|
addge01d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 0 ≤ ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ↔ ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ≤ ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) ) ) |
| 33 |
29 32
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ≤ ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) ) |
| 34 |
|
ccatlen |
⊢ ( ( ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ∧ ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) → ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) = ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) ) |
| 35 |
11 13 34
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) = ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) ) |
| 36 |
33 35
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ≤ ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) |
| 37 |
19 22 25 26 36
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ∈ ( 0 ... ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) ) |
| 38 |
18 37
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ∈ ( { ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) } × ( 0 ... ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) ) ) |
| 39 |
9 15 38
|
rspcedvdw |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ∃ 𝑤 ∈ Word 𝐴 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ∈ ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 40 |
39
|
eliund |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 41 |
40 1
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ∈ 𝑈 ) |
| 42 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) |
| 43 |
|
xp1st |
⊢ ( 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) → ( 1st ‘ 𝑏 ) ∈ { 𝑢 } ) |
| 44 |
|
elsni |
⊢ ( ( 1st ‘ 𝑏 ) ∈ { 𝑢 } → ( 1st ‘ 𝑏 ) = 𝑢 ) |
| 45 |
42 43 44
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( 1st ‘ 𝑏 ) = 𝑢 ) |
| 46 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → 𝑢 ∈ Word 𝐴 ) |
| 47 |
45 46
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( 1st ‘ 𝑏 ) ∈ Word 𝐴 ) |
| 48 |
47
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( 1st ‘ 𝑏 ) ∈ Word 𝐴 ) |
| 49 |
1
|
eleq2i |
⊢ ( 𝑏 ∈ 𝑈 ↔ 𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 50 |
49
|
biimpi |
⊢ ( 𝑏 ∈ 𝑈 → 𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 51 |
50
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → 𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 52 |
|
eliun |
⊢ ( 𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ↔ ∃ 𝑤 ∈ Word 𝐴 𝑏 ∈ ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 53 |
51 52
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → ∃ 𝑤 ∈ Word 𝐴 𝑏 ∈ ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 54 |
|
sneq |
⊢ ( 𝑢 = 𝑤 → { 𝑢 } = { 𝑤 } ) |
| 55 |
|
fveq2 |
⊢ ( 𝑢 = 𝑤 → ( ♯ ‘ 𝑢 ) = ( ♯ ‘ 𝑤 ) ) |
| 56 |
55
|
oveq2d |
⊢ ( 𝑢 = 𝑤 → ( 0 ... ( ♯ ‘ 𝑢 ) ) = ( 0 ... ( ♯ ‘ 𝑤 ) ) ) |
| 57 |
54 56
|
xpeq12d |
⊢ ( 𝑢 = 𝑤 → ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) = ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 58 |
57
|
eleq2d |
⊢ ( 𝑢 = 𝑤 → ( 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ↔ 𝑏 ∈ ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
| 59 |
58
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ Word 𝐴 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ↔ ∃ 𝑤 ∈ Word 𝐴 𝑏 ∈ ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 60 |
53 59
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → ∃ 𝑢 ∈ Word 𝐴 𝑏 ∈ ( { 𝑢 } × ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) |
| 61 |
48 60
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → ( 1st ‘ 𝑏 ) ∈ Word 𝐴 ) |
| 62 |
|
pfxcl |
⊢ ( ( 1st ‘ 𝑏 ) ∈ Word 𝐴 → ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∈ Word 𝐴 ) |
| 63 |
61 62
|
syl |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∈ Word 𝐴 ) |
| 64 |
|
swrdcl |
⊢ ( ( 1st ‘ 𝑏 ) ∈ Word 𝐴 → ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ∈ Word 𝐴 ) |
| 65 |
61 64
|
syl |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ∈ Word 𝐴 ) |
| 66 |
63 65
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) → 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ∈ ( Word 𝐴 × Word 𝐴 ) ) |
| 67 |
51
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 68 |
|
eliunxp |
⊢ ( 𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ↔ ∃ 𝑤 ∃ 𝑛 ( 𝑏 = 〈 𝑤 , 𝑛 〉 ∧ ( 𝑤 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
| 69 |
67 68
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ∃ 𝑤 ∃ 𝑛 ( 𝑏 = 〈 𝑤 , 𝑛 〉 ∧ ( 𝑤 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
| 70 |
|
opeq1 |
⊢ ( 𝑢 = 𝑤 → 〈 𝑢 , 𝑛 〉 = 〈 𝑤 , 𝑛 〉 ) |
| 71 |
70
|
eqeq2d |
⊢ ( 𝑢 = 𝑤 → ( 𝑏 = 〈 𝑢 , 𝑛 〉 ↔ 𝑏 = 〈 𝑤 , 𝑛 〉 ) ) |
| 72 |
|
eleq1w |
⊢ ( 𝑢 = 𝑤 → ( 𝑢 ∈ Word 𝐴 ↔ 𝑤 ∈ Word 𝐴 ) ) |
| 73 |
56
|
eleq2d |
⊢ ( 𝑢 = 𝑤 → ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ↔ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) |
| 74 |
72 73
|
anbi12d |
⊢ ( 𝑢 = 𝑤 → ( ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ↔ ( 𝑤 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
| 75 |
71 74
|
anbi12d |
⊢ ( 𝑢 = 𝑤 → ( ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) ↔ ( 𝑏 = 〈 𝑤 , 𝑛 〉 ∧ ( 𝑤 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
| 76 |
75
|
exbidv |
⊢ ( 𝑢 = 𝑤 → ( ∃ 𝑛 ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑛 ( 𝑏 = 〈 𝑤 , 𝑛 〉 ∧ ( 𝑤 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
| 77 |
76
|
cbvexvw |
⊢ ( ∃ 𝑢 ∃ 𝑛 ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) ↔ ∃ 𝑤 ∃ 𝑛 ( 𝑏 = 〈 𝑤 , 𝑛 〉 ∧ ( 𝑤 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
| 78 |
69 77
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ∃ 𝑢 ∃ 𝑛 ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) ) |
| 79 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) |
| 80 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) |
| 81 |
79 80
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) = ( ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ++ ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ) |
| 82 |
|
vex |
⊢ 𝑢 ∈ V |
| 83 |
|
vex |
⊢ 𝑛 ∈ V |
| 84 |
82 83
|
op1std |
⊢ ( 𝑏 = 〈 𝑢 , 𝑛 〉 → ( 1st ‘ 𝑏 ) = 𝑢 ) |
| 85 |
84
|
ad5antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 1st ‘ 𝑏 ) = 𝑢 ) |
| 86 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → 𝑢 ∈ Word 𝐴 ) |
| 87 |
85 86
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 1st ‘ 𝑏 ) ∈ Word 𝐴 ) |
| 88 |
82 83
|
op2ndd |
⊢ ( 𝑏 = 〈 𝑢 , 𝑛 〉 → ( 2nd ‘ 𝑏 ) = 𝑛 ) |
| 89 |
88
|
ad5antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 2nd ‘ 𝑏 ) = 𝑛 ) |
| 90 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) |
| 91 |
85
|
eqcomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → 𝑢 = ( 1st ‘ 𝑏 ) ) |
| 92 |
91
|
fveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( ♯ ‘ 𝑢 ) = ( ♯ ‘ ( 1st ‘ 𝑏 ) ) ) |
| 93 |
92
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 0 ... ( ♯ ‘ 𝑢 ) ) = ( 0 ... ( ♯ ‘ ( 1st ‘ 𝑏 ) ) ) ) |
| 94 |
90 93
|
eleqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → 𝑛 ∈ ( 0 ... ( ♯ ‘ ( 1st ‘ 𝑏 ) ) ) ) |
| 95 |
89 94
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 2nd ‘ 𝑏 ) ∈ ( 0 ... ( ♯ ‘ ( 1st ‘ 𝑏 ) ) ) ) |
| 96 |
|
pfxcctswrd |
⊢ ( ( ( 1st ‘ 𝑏 ) ∈ Word 𝐴 ∧ ( 2nd ‘ 𝑏 ) ∈ ( 0 ... ( ♯ ‘ ( 1st ‘ 𝑏 ) ) ) ) → ( ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ++ ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) = ( 1st ‘ 𝑏 ) ) |
| 97 |
87 95 96
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ++ ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) = ( 1st ‘ 𝑏 ) ) |
| 98 |
81 97
|
eqtr2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) |
| 99 |
79
|
fveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( ♯ ‘ ( 1st ‘ 𝑎 ) ) = ( ♯ ‘ ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ) |
| 100 |
|
pfxlen |
⊢ ( ( ( 1st ‘ 𝑏 ) ∈ Word 𝐴 ∧ ( 2nd ‘ 𝑏 ) ∈ ( 0 ... ( ♯ ‘ ( 1st ‘ 𝑏 ) ) ) ) → ( ♯ ‘ ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) = ( 2nd ‘ 𝑏 ) ) |
| 101 |
87 95 100
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( ♯ ‘ ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) = ( 2nd ‘ 𝑏 ) ) |
| 102 |
99 101
|
eqtr2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) |
| 103 |
98 102
|
jca |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) → ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) |
| 104 |
103
|
anasss |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ) → ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) |
| 105 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) |
| 106 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) |
| 107 |
105 106
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) = ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) prefix ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) |
| 108 |
11
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
| 109 |
13
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
| 110 |
|
pfxccat1 |
⊢ ( ( ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ∧ ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) → ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) prefix ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) = ( 1st ‘ 𝑎 ) ) |
| 111 |
108 109 110
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) prefix ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) = ( 1st ‘ 𝑎 ) ) |
| 112 |
107 111
|
eqtr2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ) |
| 113 |
105
|
fveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ♯ ‘ ( 1st ‘ 𝑏 ) ) = ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ) |
| 114 |
108 109 34
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ♯ ‘ ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) = ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) ) |
| 115 |
113 114
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ♯ ‘ ( 1st ‘ 𝑏 ) ) = ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) ) |
| 116 |
106 115
|
opeq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 = 〈 ( ♯ ‘ ( 1st ‘ 𝑎 ) ) , ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) 〉 ) |
| 117 |
105 116
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) = ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) substr 〈 ( ♯ ‘ ( 1st ‘ 𝑎 ) ) , ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) 〉 ) ) |
| 118 |
|
swrdccat2 |
⊢ ( ( ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ∧ ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) → ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) substr 〈 ( ♯ ‘ ( 1st ‘ 𝑎 ) ) , ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) 〉 ) = ( 2nd ‘ 𝑎 ) ) |
| 119 |
108 109 118
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) substr 〈 ( ♯ ‘ ( 1st ‘ 𝑎 ) ) , ( ( ♯ ‘ ( 1st ‘ 𝑎 ) ) + ( ♯ ‘ ( 2nd ‘ 𝑎 ) ) ) 〉 ) = ( 2nd ‘ 𝑎 ) ) |
| 120 |
117 119
|
eqtr2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) |
| 121 |
112 120
|
jca |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) → ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ) |
| 122 |
121
|
anasss |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ∧ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) → ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ) |
| 123 |
104 122
|
impbida |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ 𝑢 ∈ Word 𝐴 ) ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) |
| 124 |
123
|
anasss |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 = 〈 𝑢 , 𝑛 〉 ) ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) |
| 125 |
124
|
expl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) ) |
| 126 |
125
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) ) |
| 127 |
126
|
exlimdv |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ∃ 𝑛 ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) ) |
| 128 |
127
|
imp |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ ∃ 𝑛 ( 𝑏 = 〈 𝑢 , 𝑛 〉 ∧ ( 𝑢 ∈ Word 𝐴 ∧ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑢 ) ) ) ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) |
| 129 |
78 128
|
exlimddv |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) |
| 130 |
|
eqop |
⊢ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) → ( 𝑎 = 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ↔ ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ) ) |
| 131 |
130
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑎 = 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ↔ ( ( 1st ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) ∧ ( 2nd ‘ 𝑎 ) = ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) ) ) ) |
| 132 |
|
snssi |
⊢ ( 𝑤 ∈ Word 𝐴 → { 𝑤 } ⊆ Word 𝐴 ) |
| 133 |
132
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑤 ∈ Word 𝐴 ) → { 𝑤 } ⊆ Word 𝐴 ) |
| 134 |
|
fz0ssnn0 |
⊢ ( 0 ... ( ♯ ‘ 𝑤 ) ) ⊆ ℕ0 |
| 135 |
|
xpss12 |
⊢ ( ( { 𝑤 } ⊆ Word 𝐴 ∧ ( 0 ... ( ♯ ‘ 𝑤 ) ) ⊆ ℕ0 ) → ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ⊆ ( Word 𝐴 × ℕ0 ) ) |
| 136 |
133 134 135
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ⊆ ( Word 𝐴 × ℕ0 ) ) |
| 137 |
136
|
iunssd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ⊆ ( Word 𝐴 × ℕ0 ) ) |
| 138 |
137
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ∪ 𝑤 ∈ Word 𝐴 ( { 𝑤 } × ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ⊆ ( Word 𝐴 × ℕ0 ) ) |
| 139 |
138 67
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑏 ∈ ( Word 𝐴 × ℕ0 ) ) |
| 140 |
|
eqop |
⊢ ( 𝑏 ∈ ( Word 𝐴 × ℕ0 ) → ( 𝑏 = 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) |
| 141 |
139 140
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑏 = 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ↔ ( ( 1st ‘ 𝑏 ) = ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) ∧ ( 2nd ‘ 𝑏 ) = ( ♯ ‘ ( 1st ‘ 𝑎 ) ) ) ) ) |
| 142 |
129 131 141
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝑈 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑎 = 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ↔ 𝑏 = 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ) ) |
| 143 |
142
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) ∧ 𝑏 ∈ 𝑈 ) → ( 𝑎 = 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ↔ 𝑏 = 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ) ) |
| 144 |
143
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ∧ 𝑏 ∈ 𝑈 ) ) → ( 𝑎 = 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ↔ 𝑏 = 〈 ( ( 1st ‘ 𝑎 ) ++ ( 2nd ‘ 𝑎 ) ) , ( ♯ ‘ ( 1st ‘ 𝑎 ) ) 〉 ) ) |
| 145 |
2 41 66 144
|
f1ocnv2d |
⊢ ( 𝜑 → ( 𝐹 : ( Word 𝐴 × Word 𝐴 ) –1-1-onto→ 𝑈 ∧ ◡ 𝐹 = ( 𝑏 ∈ 𝑈 ↦ 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ) ) ) |
| 146 |
145
|
simpld |
⊢ ( 𝜑 → 𝐹 : ( Word 𝐴 × Word 𝐴 ) –1-1-onto→ 𝑈 ) |
| 147 |
145
|
simprd |
⊢ ( 𝜑 → ◡ 𝐹 = ( 𝑏 ∈ 𝑈 ↦ 〈 ( ( 1st ‘ 𝑏 ) prefix ( 2nd ‘ 𝑏 ) ) , ( ( 1st ‘ 𝑏 ) substr 〈 ( 2nd ‘ 𝑏 ) , ( ♯ ‘ ( 1st ‘ 𝑏 ) ) 〉 ) 〉 ) ) |
| 148 |
147 3
|
eqtr4di |
⊢ ( 𝜑 → ◡ 𝐹 = 𝐺 ) |
| 149 |
146 148
|
jca |
⊢ ( 𝜑 → ( 𝐹 : ( Word 𝐴 × Word 𝐴 ) –1-1-onto→ 𝑈 ∧ ◡ 𝐹 = 𝐺 ) ) |