Step |
Hyp |
Ref |
Expression |
1 |
|
lmlim.j |
⊢ 𝐽 ∈ ( TopOn ‘ 𝑌 ) |
2 |
|
lmlim.f |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ 𝑋 ) |
3 |
|
lmlim.p |
⊢ ( 𝜑 → 𝑃 ∈ 𝑋 ) |
4 |
|
lmlim.t |
⊢ ( 𝐽 ↾t 𝑋 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) |
5 |
|
lmlim.x |
⊢ 𝑋 ⊆ ℂ |
6 |
|
eqid |
⊢ ( 𝐽 ↾t 𝑋 ) = ( 𝐽 ↾t 𝑋 ) |
7 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
8 |
|
cnex |
⊢ ℂ ∈ V |
9 |
8
|
a1i |
⊢ ( 𝜑 → ℂ ∈ V ) |
10 |
5
|
a1i |
⊢ ( 𝜑 → 𝑋 ⊆ ℂ ) |
11 |
9 10
|
ssexd |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
12 |
1
|
topontopi |
⊢ 𝐽 ∈ Top |
13 |
12
|
a1i |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
14 |
|
1z |
⊢ 1 ∈ ℤ |
15 |
14
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
16 |
6 7 11 13 3 15 2
|
lmss |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ 𝐹 ( ⇝𝑡 ‘ ( 𝐽 ↾t 𝑋 ) ) 𝑃 ) ) |
17 |
4
|
fveq2i |
⊢ ( ⇝𝑡 ‘ ( 𝐽 ↾t 𝑋 ) ) = ( ⇝𝑡 ‘ ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) ) |
18 |
17
|
breqi |
⊢ ( 𝐹 ( ⇝𝑡 ‘ ( 𝐽 ↾t 𝑋 ) ) 𝑃 ↔ 𝐹 ( ⇝𝑡 ‘ ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) ) 𝑃 ) |
19 |
18
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( 𝐽 ↾t 𝑋 ) ) 𝑃 ↔ 𝐹 ( ⇝𝑡 ‘ ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) ) 𝑃 ) ) |
20 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) |
21 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
22 |
21
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
23 |
22
|
a1i |
⊢ ( 𝜑 → ( TopOpen ‘ ℂfld ) ∈ Top ) |
24 |
20 7 11 23 3 15 2
|
lmss |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) 𝑃 ↔ 𝐹 ( ⇝𝑡 ‘ ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) ) 𝑃 ) ) |
25 |
|
fss |
⊢ ( ( 𝐹 : ℕ ⟶ 𝑋 ∧ 𝑋 ⊆ ℂ ) → 𝐹 : ℕ ⟶ ℂ ) |
26 |
2 5 25
|
sylancl |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ℂ ) |
27 |
21 7
|
lmclimf |
⊢ ( ( 1 ∈ ℤ ∧ 𝐹 : ℕ ⟶ ℂ ) → ( 𝐹 ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) 𝑃 ↔ 𝐹 ⇝ 𝑃 ) ) |
28 |
14 26 27
|
sylancr |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( TopOpen ‘ ℂfld ) ) 𝑃 ↔ 𝐹 ⇝ 𝑃 ) ) |
29 |
24 28
|
bitr3d |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ( TopOpen ‘ ℂfld ) ↾t 𝑋 ) ) 𝑃 ↔ 𝐹 ⇝ 𝑃 ) ) |
30 |
16 19 29
|
3bitrd |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ 𝐹 ⇝ 𝑃 ) ) |