Step |
Hyp |
Ref |
Expression |
1 |
|
ulmres.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
ulmres.w |
⊢ 𝑊 = ( ℤ≥ ‘ 𝑁 ) |
3 |
|
ulmres.m |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
4 |
|
ulmres.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
5 |
|
ulmscl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V ) |
6 |
|
ulmcl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ ) |
7 |
5 6
|
jca |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) ) |
9 |
|
ulmscl |
⊢ ( ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V ) |
10 |
|
ulmcl |
⊢ ( ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ ) |
11 |
9 10
|
jca |
⊢ ( ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) ) |
13 |
3 1
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
15 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
16 |
14 15
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝑀 ∈ ℤ ) |
17 |
1
|
rexuz3 |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
18 |
16 17
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
19 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
20 |
14 19
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝑁 ∈ ℤ ) |
21 |
2
|
rexuz3 |
⊢ ( 𝑁 ∈ ℤ → ( ∃ 𝑗 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
22 |
20 21
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( ∃ 𝑗 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
23 |
18 22
|
bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ↔ ∃ 𝑗 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
24 |
23
|
ralbidv |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( ∀ 𝑟 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ↔ ∀ 𝑟 ∈ ℝ+ ∃ 𝑗 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
25 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |
26 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) ∧ ( 𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) |
27 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
28 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝐺 : 𝑆 ⟶ ℂ ) |
29 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝑆 ∈ V ) |
30 |
1 16 25 26 27 28 29
|
ulm2 |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ∀ 𝑟 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
31 |
|
uzss |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
32 |
14 31
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
33 |
32 2 1
|
3sstr4g |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → 𝑊 ⊆ 𝑍 ) |
34 |
25 33
|
fssresd |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( 𝐹 ↾ 𝑊 ) : 𝑊 ⟶ ( ℂ ↑m 𝑆 ) ) |
35 |
|
fvres |
⊢ ( 𝑘 ∈ 𝑊 → ( ( 𝐹 ↾ 𝑊 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
36 |
35
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) ∧ ( 𝑘 ∈ 𝑊 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ↾ 𝑊 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
37 |
36
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) ∧ ( 𝑘 ∈ 𝑊 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( ( 𝐹 ↾ 𝑊 ) ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) ) |
38 |
2 20 34 37 27 28 29
|
ulm2 |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ∀ 𝑟 ∈ ℝ+ ∃ 𝑗 ∈ 𝑊 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑟 ) ) |
39 |
24 30 38
|
3bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) ) → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) ) |
40 |
39
|
ex |
⊢ ( 𝜑 → ( ( 𝑆 ∈ V ∧ 𝐺 : 𝑆 ⟶ ℂ ) → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) ) ) |
41 |
8 12 40
|
pm5.21ndd |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ( 𝐹 ↾ 𝑊 ) ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) ) |