Step |
Hyp |
Ref |
Expression |
1 |
|
ulmclm.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
ulmclm.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
ulmclm.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
4 |
|
ulmclm.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
5 |
|
ulmclm.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝑊 ) |
6 |
|
ulmclm.e |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) = ( 𝐻 ‘ 𝑘 ) ) |
7 |
|
ulmclm.u |
⊢ ( 𝜑 → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) |
8 |
|
fveq2 |
⊢ ( 𝑧 = 𝐴 → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) ) |
9 |
|
fveq2 |
⊢ ( 𝑧 = 𝐴 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝐴 ) ) |
10 |
8 9
|
oveq12d |
⊢ ( 𝑧 = 𝐴 → ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) = ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) |
11 |
10
|
fveq2d |
⊢ ( 𝑧 = 𝐴 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) = ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) ) |
12 |
11
|
breq1d |
⊢ ( 𝑧 = 𝐴 → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
13 |
12
|
rspcv |
⊢ ( 𝐴 ∈ 𝑆 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
14 |
4 13
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
15 |
14
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
16 |
15
|
reximdv |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
17 |
16
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
18 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) |
19 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
20 |
|
ulmcl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ ) |
21 |
7 20
|
syl |
⊢ ( 𝜑 → 𝐺 : 𝑆 ⟶ ℂ ) |
22 |
|
ulmscl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V ) |
23 |
7 22
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ V ) |
24 |
1 2 3 18 19 21 23
|
ulm2 |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
25 |
6
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) ) |
26 |
21 4
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐴 ) ∈ ℂ ) |
27 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) ) |
28 |
|
elmapi |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
29 |
27 28
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ ) |
30 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐴 ∈ 𝑆 ) |
31 |
29 30
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) ∈ ℂ ) |
32 |
1 2 5 25 26 31
|
clim2c |
⊢ ( 𝜑 → ( 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) ) |
33 |
17 24 32
|
3imtr4d |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) ) ) |
34 |
7 33
|
mpd |
⊢ ( 𝜑 → 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) ) |