Step |
Hyp |
Ref |
Expression |
1 |
|
simpl1 |
⊢ ( ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → 𝑆 ∈ 𝐴 ) |
2 |
|
simpl2 |
⊢ ( ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → 𝑁 ∈ ℕ0 ) |
3 |
|
simpr |
⊢ ( ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → 𝑥 ∈ ( 0 ..^ 𝑁 ) ) |
4 |
|
repswsymb |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑆 repeatS 𝑁 ) ‘ 𝑥 ) = 𝑆 ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑆 repeatS 𝑁 ) ‘ 𝑥 ) = 𝑆 ) |
6 |
5
|
fveq2d |
⊢ ( ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝐹 ‘ ( ( 𝑆 repeatS 𝑁 ) ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑆 ) ) |
7 |
6
|
mpteq2dva |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝐹 ‘ ( ( 𝑆 repeatS 𝑁 ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝐹 ‘ 𝑆 ) ) ) |
8 |
|
simp3 |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
9 |
|
repsf |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑆 repeatS 𝑁 ) : ( 0 ..^ 𝑁 ) ⟶ 𝐴 ) |
10 |
9
|
3adant3 |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( 𝑆 repeatS 𝑁 ) : ( 0 ..^ 𝑁 ) ⟶ 𝐴 ) |
11 |
|
fcompt |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝑆 repeatS 𝑁 ) : ( 0 ..^ 𝑁 ) ⟶ 𝐴 ) → ( 𝐹 ∘ ( 𝑆 repeatS 𝑁 ) ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝐹 ‘ ( ( 𝑆 repeatS 𝑁 ) ‘ 𝑥 ) ) ) ) |
12 |
8 10 11
|
syl2anc |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( 𝐹 ∘ ( 𝑆 repeatS 𝑁 ) ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝐹 ‘ ( ( 𝑆 repeatS 𝑁 ) ‘ 𝑥 ) ) ) ) |
13 |
|
fvexd |
⊢ ( 𝑆 ∈ 𝐴 → ( 𝐹 ‘ 𝑆 ) ∈ V ) |
14 |
13
|
anim1i |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐹 ‘ 𝑆 ) ∈ V ∧ 𝑁 ∈ ℕ0 ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( ( 𝐹 ‘ 𝑆 ) ∈ V ∧ 𝑁 ∈ ℕ0 ) ) |
16 |
|
reps |
⊢ ( ( ( 𝐹 ‘ 𝑆 ) ∈ V ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐹 ‘ 𝑆 ) repeatS 𝑁 ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝐹 ‘ 𝑆 ) ) ) |
17 |
15 16
|
syl |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( ( 𝐹 ‘ 𝑆 ) repeatS 𝑁 ) = ( 𝑥 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝐹 ‘ 𝑆 ) ) ) |
18 |
7 12 17
|
3eqtr4d |
⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( 𝐹 ∘ ( 𝑆 repeatS 𝑁 ) ) = ( ( 𝐹 ‘ 𝑆 ) repeatS 𝑁 ) ) |