Step |
Hyp |
Ref |
Expression |
1 |
|
swrdccatind.l |
⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = 𝐿 ) |
2 |
|
swrdccatind.w |
⊢ ( 𝜑 → ( 𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ) ) |
3 |
|
pfxccatin12d.m |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝐿 ) ) |
4 |
|
pfxccatin12d.n |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝐿 ... ( 𝐿 + ( ♯ ‘ 𝐵 ) ) ) ) |
5 |
1
|
oveq2d |
⊢ ( 𝜑 → ( 0 ... ( ♯ ‘ 𝐴 ) ) = ( 0 ... 𝐿 ) ) |
6 |
5
|
eleq2d |
⊢ ( 𝜑 → ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝐴 ) ) ↔ 𝑀 ∈ ( 0 ... 𝐿 ) ) ) |
7 |
1
|
oveq1d |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) = ( 𝐿 + ( ♯ ‘ 𝐵 ) ) ) |
8 |
1 7
|
oveq12d |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) ... ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) = ( 𝐿 ... ( 𝐿 + ( ♯ ‘ 𝐵 ) ) ) ) |
9 |
8
|
eleq2d |
⊢ ( 𝜑 → ( 𝑁 ∈ ( ( ♯ ‘ 𝐴 ) ... ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ↔ 𝑁 ∈ ( 𝐿 ... ( 𝐿 + ( ♯ ‘ 𝐵 ) ) ) ) ) |
10 |
6 9
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝐴 ) ) ∧ 𝑁 ∈ ( ( ♯ ‘ 𝐴 ) ... ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ) ↔ ( 𝑀 ∈ ( 0 ... 𝐿 ) ∧ 𝑁 ∈ ( 𝐿 ... ( 𝐿 + ( ♯ ‘ 𝐵 ) ) ) ) ) ) |
11 |
3 4 10
|
mpbir2and |
⊢ ( 𝜑 → ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝐴 ) ) ∧ 𝑁 ∈ ( ( ♯ ‘ 𝐴 ) ... ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ) ) |
12 |
|
eqid |
⊢ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐴 ) |
13 |
12
|
pfxccatin12 |
⊢ ( ( 𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ) → ( ( 𝑀 ∈ ( 0 ... ( ♯ ‘ 𝐴 ) ) ∧ 𝑁 ∈ ( ( ♯ ‘ 𝐴 ) ... ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ 𝐵 ) ) ) ) → ( ( 𝐴 ++ 𝐵 ) substr 〈 𝑀 , 𝑁 〉 ) = ( ( 𝐴 substr 〈 𝑀 , ( ♯ ‘ 𝐴 ) 〉 ) ++ ( 𝐵 prefix ( 𝑁 − ( ♯ ‘ 𝐴 ) ) ) ) ) ) |
14 |
2 11 13
|
sylc |
⊢ ( 𝜑 → ( ( 𝐴 ++ 𝐵 ) substr 〈 𝑀 , 𝑁 〉 ) = ( ( 𝐴 substr 〈 𝑀 , ( ♯ ‘ 𝐴 ) 〉 ) ++ ( 𝐵 prefix ( 𝑁 − ( ♯ ‘ 𝐴 ) ) ) ) ) |
15 |
1
|
opeq2d |
⊢ ( 𝜑 → 〈 𝑀 , ( ♯ ‘ 𝐴 ) 〉 = 〈 𝑀 , 𝐿 〉 ) |
16 |
15
|
oveq2d |
⊢ ( 𝜑 → ( 𝐴 substr 〈 𝑀 , ( ♯ ‘ 𝐴 ) 〉 ) = ( 𝐴 substr 〈 𝑀 , 𝐿 〉 ) ) |
17 |
1
|
oveq2d |
⊢ ( 𝜑 → ( 𝑁 − ( ♯ ‘ 𝐴 ) ) = ( 𝑁 − 𝐿 ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝜑 → ( 𝐵 prefix ( 𝑁 − ( ♯ ‘ 𝐴 ) ) ) = ( 𝐵 prefix ( 𝑁 − 𝐿 ) ) ) |
19 |
16 18
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝐴 substr 〈 𝑀 , ( ♯ ‘ 𝐴 ) 〉 ) ++ ( 𝐵 prefix ( 𝑁 − ( ♯ ‘ 𝐴 ) ) ) ) = ( ( 𝐴 substr 〈 𝑀 , 𝐿 〉 ) ++ ( 𝐵 prefix ( 𝑁 − 𝐿 ) ) ) ) |
20 |
14 19
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐴 ++ 𝐵 ) substr 〈 𝑀 , 𝑁 〉 ) = ( ( 𝐴 substr 〈 𝑀 , 𝐿 〉 ) ++ ( 𝐵 prefix ( 𝑁 − 𝐿 ) ) ) ) |