Step |
Hyp |
Ref |
Expression |
1 |
|
inss1 |
⊢ ( dom 𝐹 ∩ 𝐵 ) ⊆ dom 𝐹 |
2 |
|
ssralv |
⊢ ( ( dom 𝐹 ∩ 𝐵 ) ⊆ dom 𝐹 → ( ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) |
4 |
3
|
reximi |
⊢ ( ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) |
5 |
4
|
ralimi |
⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) |
6 |
5
|
anim2i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) |
7 |
6
|
a1i |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) → ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
8 |
|
rlimf |
⊢ ( 𝐹 ⇝𝑟 𝐴 → 𝐹 : dom 𝐹 ⟶ ℂ ) |
9 |
|
rlimss |
⊢ ( 𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ ) |
10 |
|
eqidd |
⊢ ( ( 𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
11 |
8 9 10
|
rlim |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ⇝𝑟 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom 𝐹 ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
12 |
|
fssres |
⊢ ( ( 𝐹 : dom 𝐹 ⟶ ℂ ∧ ( dom 𝐹 ∩ 𝐵 ) ⊆ dom 𝐹 ) → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ) |
13 |
8 1 12
|
sylancl |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ) |
14 |
|
resres |
⊢ ( ( 𝐹 ↾ dom 𝐹 ) ↾ 𝐵 ) = ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) |
15 |
|
ffn |
⊢ ( 𝐹 : dom 𝐹 ⟶ ℂ → 𝐹 Fn dom 𝐹 ) |
16 |
|
fnresdm |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 ) |
17 |
8 15 16
|
3syl |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 ) |
18 |
17
|
reseq1d |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐹 ↾ dom 𝐹 ) ↾ 𝐵 ) = ( 𝐹 ↾ 𝐵 ) ) |
19 |
14 18
|
eqtr3id |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) = ( 𝐹 ↾ 𝐵 ) ) |
20 |
19
|
feq1d |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐹 ↾ ( dom 𝐹 ∩ 𝐵 ) ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ↔ ( 𝐹 ↾ 𝐵 ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ) ) |
21 |
13 20
|
mpbid |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) : ( dom 𝐹 ∩ 𝐵 ) ⟶ ℂ ) |
22 |
1 9
|
sstrid |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( dom 𝐹 ∩ 𝐵 ) ⊆ ℝ ) |
23 |
|
elinel2 |
⊢ ( 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) → 𝑧 ∈ 𝐵 ) |
24 |
23
|
fvresd |
⊢ ( 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ) → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
26 |
21 22 25
|
rlim |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 ↔ ( 𝐴 ∈ ℂ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ ( dom 𝐹 ∩ 𝐵 ) ( 𝑦 ≤ 𝑧 → ( abs ‘ ( ( 𝐹 ‘ 𝑧 ) − 𝐴 ) ) < 𝑥 ) ) ) ) |
27 |
7 11 26
|
3imtr4d |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 ) ) |
28 |
27
|
pm2.43i |
⊢ ( 𝐹 ⇝𝑟 𝐴 → ( 𝐹 ↾ 𝐵 ) ⇝𝑟 𝐴 ) |