Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝑊 ∈ Word 𝑉 ) |
2 |
1
|
anim1i |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
3 |
|
3anass |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ) |
4 |
2 3
|
sylibr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) |
5 |
|
ccatw2s1ccatws2OLD |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( ( 𝑊 ++ 〈“ 𝑋 ”〉 ) ++ 〈“ 𝑌 ”〉 ) = ( 𝑊 ++ 〈“ 𝑋 𝑌 ”〉 ) ) |
6 |
5
|
fveq1d |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( ( ( 𝑊 ++ 〈“ 𝑋 ”〉 ) ++ 〈“ 𝑌 ”〉 ) ‘ 𝐼 ) = ( ( 𝑊 ++ 〈“ 𝑋 𝑌 ”〉 ) ‘ 𝐼 ) ) |
7 |
4 6
|
syl |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( 𝑊 ++ 〈“ 𝑋 ”〉 ) ++ 〈“ 𝑌 ”〉 ) ‘ 𝐼 ) = ( ( 𝑊 ++ 〈“ 𝑋 𝑌 ”〉 ) ‘ 𝐼 ) ) |
8 |
1
|
adantr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑊 ∈ Word 𝑉 ) |
9 |
|
s2cl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → 〈“ 𝑋 𝑌 ”〉 ∈ Word 𝑉 ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 〈“ 𝑋 𝑌 ”〉 ∈ Word 𝑉 ) |
11 |
|
simp2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝐼 ∈ ℕ0 ) |
12 |
|
lencl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
13 |
12
|
nn0zd |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
14 |
13
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
15 |
|
simp3 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝐼 < ( ♯ ‘ 𝑊 ) ) |
16 |
|
elfzo0z |
⊢ ( 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( 𝐼 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ) |
17 |
11 14 15 16
|
syl3anbrc |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) → 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
19 |
|
ccatval1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 〈“ 𝑋 𝑌 ”〉 ∈ Word 𝑉 ∧ 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 ++ 〈“ 𝑋 𝑌 ”〉 ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) ) |
20 |
8 10 18 19
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝑊 ++ 〈“ 𝑋 𝑌 ”〉 ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) ) |
21 |
7 20
|
eqtrd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < ( ♯ ‘ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( 𝑊 ++ 〈“ 𝑋 ”〉 ) ++ 〈“ 𝑌 ”〉 ) ‘ 𝐼 ) = ( 𝑊 ‘ 𝐼 ) ) |