Step |
Hyp |
Ref |
Expression |
1 |
|
splval2.a |
⊢ ( 𝜑 → 𝐴 ∈ Word 𝑋 ) |
2 |
|
splval2.b |
⊢ ( 𝜑 → 𝐵 ∈ Word 𝑋 ) |
3 |
|
splval2.c |
⊢ ( 𝜑 → 𝐶 ∈ Word 𝑋 ) |
4 |
|
splval2.r |
⊢ ( 𝜑 → 𝑅 ∈ Word 𝑋 ) |
5 |
|
splval2.s |
⊢ ( 𝜑 → 𝑆 = ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ) |
6 |
|
splval2.f |
⊢ ( 𝜑 → 𝐹 = ( ♯ ‘ 𝐴 ) ) |
7 |
|
splval2.t |
⊢ ( 𝜑 → 𝑇 = ( 𝐹 + ( ♯ ‘ 𝐵 ) ) ) |
8 |
|
ccatcl |
⊢ ( ( 𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋 ) → ( 𝐴 ++ 𝐵 ) ∈ Word 𝑋 ) |
9 |
1 2 8
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ++ 𝐵 ) ∈ Word 𝑋 ) |
10 |
|
ccatcl |
⊢ ( ( ( 𝐴 ++ 𝐵 ) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋 ) → ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ∈ Word 𝑋 ) |
11 |
9 3 10
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ∈ Word 𝑋 ) |
12 |
5 11
|
eqeltrd |
⊢ ( 𝜑 → 𝑆 ∈ Word 𝑋 ) |
13 |
|
lencl |
⊢ ( 𝐴 ∈ Word 𝑋 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 ) |
14 |
1 13
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 ) |
15 |
6 14
|
eqeltrd |
⊢ ( 𝜑 → 𝐹 ∈ ℕ0 ) |
16 |
|
lencl |
⊢ ( 𝐵 ∈ Word 𝑋 → ( ♯ ‘ 𝐵 ) ∈ ℕ0 ) |
17 |
2 16
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℕ0 ) |
18 |
15 17
|
nn0addcld |
⊢ ( 𝜑 → ( 𝐹 + ( ♯ ‘ 𝐵 ) ) ∈ ℕ0 ) |
19 |
7 18
|
eqeltrd |
⊢ ( 𝜑 → 𝑇 ∈ ℕ0 ) |
20 |
|
splval |
⊢ ( ( 𝑆 ∈ Word 𝑋 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝑇 ∈ ℕ0 ∧ 𝑅 ∈ Word 𝑋 ) ) → ( 𝑆 splice 〈 𝐹 , 𝑇 , 𝑅 〉 ) = ( ( ( 𝑆 prefix 𝐹 ) ++ 𝑅 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) ) |
21 |
12 15 19 4 20
|
syl13anc |
⊢ ( 𝜑 → ( 𝑆 splice 〈 𝐹 , 𝑇 , 𝑅 〉 ) = ( ( ( 𝑆 prefix 𝐹 ) ++ 𝑅 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) ) |
22 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
23 |
15 22
|
eleqtrdi |
⊢ ( 𝜑 → 𝐹 ∈ ( ℤ≥ ‘ 0 ) ) |
24 |
15
|
nn0zd |
⊢ ( 𝜑 → 𝐹 ∈ ℤ ) |
25 |
24
|
uzidd |
⊢ ( 𝜑 → 𝐹 ∈ ( ℤ≥ ‘ 𝐹 ) ) |
26 |
|
uzaddcl |
⊢ ( ( 𝐹 ∈ ( ℤ≥ ‘ 𝐹 ) ∧ ( ♯ ‘ 𝐵 ) ∈ ℕ0 ) → ( 𝐹 + ( ♯ ‘ 𝐵 ) ) ∈ ( ℤ≥ ‘ 𝐹 ) ) |
27 |
25 17 26
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 + ( ♯ ‘ 𝐵 ) ) ∈ ( ℤ≥ ‘ 𝐹 ) ) |
28 |
7 27
|
eqeltrd |
⊢ ( 𝜑 → 𝑇 ∈ ( ℤ≥ ‘ 𝐹 ) ) |
29 |
|
elfzuzb |
⊢ ( 𝐹 ∈ ( 0 ... 𝑇 ) ↔ ( 𝐹 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑇 ∈ ( ℤ≥ ‘ 𝐹 ) ) ) |
30 |
23 28 29
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 ∈ ( 0 ... 𝑇 ) ) |
31 |
19 22
|
eleqtrdi |
⊢ ( 𝜑 → 𝑇 ∈ ( ℤ≥ ‘ 0 ) ) |
32 |
|
ccatlen |
⊢ ( ( ( 𝐴 ++ 𝐵 ) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋 ) → ( ♯ ‘ ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ) = ( ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) + ( ♯ ‘ 𝐶 ) ) ) |
33 |
9 3 32
|
syl2anc |
⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ) = ( ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) + ( ♯ ‘ 𝐶 ) ) ) |
34 |
5
|
fveq2d |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) = ( ♯ ‘ ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ) ) |
35 |
6
|
oveq1d |
⊢ ( 𝜑 → ( 𝐹 + ( ♯ ‘ 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) |
36 |
|
ccatlen |
⊢ ( ( 𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋 ) → ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) |
37 |
1 2 36
|
syl2anc |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) = ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) |
38 |
35 7 37
|
3eqtr4d |
⊢ ( 𝜑 → 𝑇 = ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) ) |
39 |
38
|
oveq1d |
⊢ ( 𝜑 → ( 𝑇 + ( ♯ ‘ 𝐶 ) ) = ( ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) + ( ♯ ‘ 𝐶 ) ) ) |
40 |
33 34 39
|
3eqtr4d |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) = ( 𝑇 + ( ♯ ‘ 𝐶 ) ) ) |
41 |
19
|
nn0zd |
⊢ ( 𝜑 → 𝑇 ∈ ℤ ) |
42 |
41
|
uzidd |
⊢ ( 𝜑 → 𝑇 ∈ ( ℤ≥ ‘ 𝑇 ) ) |
43 |
|
lencl |
⊢ ( 𝐶 ∈ Word 𝑋 → ( ♯ ‘ 𝐶 ) ∈ ℕ0 ) |
44 |
3 43
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐶 ) ∈ ℕ0 ) |
45 |
|
uzaddcl |
⊢ ( ( 𝑇 ∈ ( ℤ≥ ‘ 𝑇 ) ∧ ( ♯ ‘ 𝐶 ) ∈ ℕ0 ) → ( 𝑇 + ( ♯ ‘ 𝐶 ) ) ∈ ( ℤ≥ ‘ 𝑇 ) ) |
46 |
42 44 45
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 + ( ♯ ‘ 𝐶 ) ) ∈ ( ℤ≥ ‘ 𝑇 ) ) |
47 |
40 46
|
eqeltrd |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝑇 ) ) |
48 |
|
elfzuzb |
⊢ ( 𝑇 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ↔ ( 𝑇 ∈ ( ℤ≥ ‘ 0 ) ∧ ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝑇 ) ) ) |
49 |
31 47 48
|
sylanbrc |
⊢ ( 𝜑 → 𝑇 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
50 |
|
ccatpfx |
⊢ ( ( 𝑆 ∈ Word 𝑋 ∧ 𝐹 ∈ ( 0 ... 𝑇 ) ∧ 𝑇 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ( 𝑆 prefix 𝐹 ) ++ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ) = ( 𝑆 prefix 𝑇 ) ) |
51 |
12 30 49 50
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝐹 ) ++ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ) = ( 𝑆 prefix 𝑇 ) ) |
52 |
|
lencl |
⊢ ( 𝑆 ∈ Word 𝑋 → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
53 |
12 52
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
54 |
53 22
|
eleqtrdi |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 0 ) ) |
55 |
|
eluzfz2 |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 0 ) → ( ♯ ‘ 𝑆 ) ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
56 |
54 55
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
57 |
|
ccatpfx |
⊢ ( ( 𝑆 ∈ Word 𝑋 ∧ 𝑇 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ∧ ( ♯ ‘ 𝑆 ) ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ( 𝑆 prefix 𝑇 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) = ( 𝑆 prefix ( ♯ ‘ 𝑆 ) ) ) |
58 |
12 49 56 57
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝑇 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) = ( 𝑆 prefix ( ♯ ‘ 𝑆 ) ) ) |
59 |
|
pfxid |
⊢ ( 𝑆 ∈ Word 𝑋 → ( 𝑆 prefix ( ♯ ‘ 𝑆 ) ) = 𝑆 ) |
60 |
12 59
|
syl |
⊢ ( 𝜑 → ( 𝑆 prefix ( ♯ ‘ 𝑆 ) ) = 𝑆 ) |
61 |
58 60 5
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝑇 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) = ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ) |
62 |
|
pfxcl |
⊢ ( 𝑆 ∈ Word 𝑋 → ( 𝑆 prefix 𝑇 ) ∈ Word 𝑋 ) |
63 |
12 62
|
syl |
⊢ ( 𝜑 → ( 𝑆 prefix 𝑇 ) ∈ Word 𝑋 ) |
64 |
|
swrdcl |
⊢ ( 𝑆 ∈ Word 𝑋 → ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ∈ Word 𝑋 ) |
65 |
12 64
|
syl |
⊢ ( 𝜑 → ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ∈ Word 𝑋 ) |
66 |
|
pfxlen |
⊢ ( ( 𝑆 ∈ Word 𝑋 ∧ 𝑇 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ ( 𝑆 prefix 𝑇 ) ) = 𝑇 ) |
67 |
12 49 66
|
syl2anc |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑆 prefix 𝑇 ) ) = 𝑇 ) |
68 |
67 38
|
eqtrd |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑆 prefix 𝑇 ) ) = ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) ) |
69 |
|
ccatopth |
⊢ ( ( ( ( 𝑆 prefix 𝑇 ) ∈ Word 𝑋 ∧ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ∈ Word 𝑋 ) ∧ ( ( 𝐴 ++ 𝐵 ) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋 ) ∧ ( ♯ ‘ ( 𝑆 prefix 𝑇 ) ) = ( ♯ ‘ ( 𝐴 ++ 𝐵 ) ) ) → ( ( ( 𝑆 prefix 𝑇 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) = ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ↔ ( ( 𝑆 prefix 𝑇 ) = ( 𝐴 ++ 𝐵 ) ∧ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) = 𝐶 ) ) ) |
70 |
63 65 9 3 68 69
|
syl221anc |
⊢ ( 𝜑 → ( ( ( 𝑆 prefix 𝑇 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) = ( ( 𝐴 ++ 𝐵 ) ++ 𝐶 ) ↔ ( ( 𝑆 prefix 𝑇 ) = ( 𝐴 ++ 𝐵 ) ∧ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) = 𝐶 ) ) ) |
71 |
61 70
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝑇 ) = ( 𝐴 ++ 𝐵 ) ∧ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) = 𝐶 ) ) |
72 |
71
|
simpld |
⊢ ( 𝜑 → ( 𝑆 prefix 𝑇 ) = ( 𝐴 ++ 𝐵 ) ) |
73 |
51 72
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝐹 ) ++ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ) = ( 𝐴 ++ 𝐵 ) ) |
74 |
|
pfxcl |
⊢ ( 𝑆 ∈ Word 𝑋 → ( 𝑆 prefix 𝐹 ) ∈ Word 𝑋 ) |
75 |
12 74
|
syl |
⊢ ( 𝜑 → ( 𝑆 prefix 𝐹 ) ∈ Word 𝑋 ) |
76 |
|
swrdcl |
⊢ ( 𝑆 ∈ Word 𝑋 → ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ∈ Word 𝑋 ) |
77 |
12 76
|
syl |
⊢ ( 𝜑 → ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ∈ Word 𝑋 ) |
78 |
|
uztrn |
⊢ ( ( ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝑇 ) ∧ 𝑇 ∈ ( ℤ≥ ‘ 𝐹 ) ) → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝐹 ) ) |
79 |
47 28 78
|
syl2anc |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝐹 ) ) |
80 |
|
elfzuzb |
⊢ ( 𝐹 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ↔ ( 𝐹 ∈ ( ℤ≥ ‘ 0 ) ∧ ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝐹 ) ) ) |
81 |
23 79 80
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
82 |
|
pfxlen |
⊢ ( ( 𝑆 ∈ Word 𝑋 ∧ 𝐹 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ ( 𝑆 prefix 𝐹 ) ) = 𝐹 ) |
83 |
12 81 82
|
syl2anc |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑆 prefix 𝐹 ) ) = 𝐹 ) |
84 |
83 6
|
eqtrd |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑆 prefix 𝐹 ) ) = ( ♯ ‘ 𝐴 ) ) |
85 |
|
ccatopth |
⊢ ( ( ( ( 𝑆 prefix 𝐹 ) ∈ Word 𝑋 ∧ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ∈ Word 𝑋 ) ∧ ( 𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋 ) ∧ ( ♯ ‘ ( 𝑆 prefix 𝐹 ) ) = ( ♯ ‘ 𝐴 ) ) → ( ( ( 𝑆 prefix 𝐹 ) ++ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ) = ( 𝐴 ++ 𝐵 ) ↔ ( ( 𝑆 prefix 𝐹 ) = 𝐴 ∧ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) = 𝐵 ) ) ) |
86 |
75 77 1 2 84 85
|
syl221anc |
⊢ ( 𝜑 → ( ( ( 𝑆 prefix 𝐹 ) ++ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) ) = ( 𝐴 ++ 𝐵 ) ↔ ( ( 𝑆 prefix 𝐹 ) = 𝐴 ∧ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) = 𝐵 ) ) ) |
87 |
73 86
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝐹 ) = 𝐴 ∧ ( 𝑆 substr 〈 𝐹 , 𝑇 〉 ) = 𝐵 ) ) |
88 |
87
|
simpld |
⊢ ( 𝜑 → ( 𝑆 prefix 𝐹 ) = 𝐴 ) |
89 |
88
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑆 prefix 𝐹 ) ++ 𝑅 ) = ( 𝐴 ++ 𝑅 ) ) |
90 |
71
|
simprd |
⊢ ( 𝜑 → ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) = 𝐶 ) |
91 |
89 90
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 𝑆 prefix 𝐹 ) ++ 𝑅 ) ++ ( 𝑆 substr 〈 𝑇 , ( ♯ ‘ 𝑆 ) 〉 ) ) = ( ( 𝐴 ++ 𝑅 ) ++ 𝐶 ) ) |
92 |
21 91
|
eqtrd |
⊢ ( 𝜑 → ( 𝑆 splice 〈 𝐹 , 𝑇 , 𝑅 〉 ) = ( ( 𝐴 ++ 𝑅 ) ++ 𝐶 ) ) |