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 ) ) |