Step |
Hyp |
Ref |
Expression |
1 |
|
mbfdm |
⊢ ( 𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol ) |
2 |
|
fdm |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → dom 𝐹 = 𝐴 ) |
3 |
2
|
eleq1d |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( dom 𝐹 ∈ dom vol ↔ 𝐴 ∈ dom vol ) ) |
4 |
1 3
|
syl5ib |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( 𝐹 ∈ MblFn → 𝐴 ∈ dom vol ) ) |
5 |
|
mbfdm |
⊢ ( ( ℜ ∘ 𝐹 ) ∈ MblFn → dom ( ℜ ∘ 𝐹 ) ∈ dom vol ) |
6 |
5
|
adantr |
⊢ ( ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) → dom ( ℜ ∘ 𝐹 ) ∈ dom vol ) |
7 |
|
ref |
⊢ ℜ : ℂ ⟶ ℝ |
8 |
|
fco |
⊢ ( ( ℜ : ℂ ⟶ ℝ ∧ 𝐹 : 𝐴 ⟶ ℂ ) → ( ℜ ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
9 |
7 8
|
mpan |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( ℜ ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
10 |
9
|
fdmd |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → dom ( ℜ ∘ 𝐹 ) = 𝐴 ) |
11 |
10
|
eleq1d |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( dom ( ℜ ∘ 𝐹 ) ∈ dom vol ↔ 𝐴 ∈ dom vol ) ) |
12 |
6 11
|
syl5ib |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) → 𝐴 ∈ dom vol ) ) |
13 |
|
ismbf1 |
⊢ ( 𝐹 ∈ MblFn ↔ ( 𝐹 ∈ ( ℂ ↑pm ℝ ) ∧ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) |
14 |
9
|
adantr |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ℜ ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
15 |
|
ismbf |
⊢ ( ( ℜ ∘ 𝐹 ) : 𝐴 ⟶ ℝ → ( ( ℜ ∘ 𝐹 ) ∈ MblFn ↔ ∀ 𝑥 ∈ ran (,) ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
16 |
14 15
|
syl |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ( ℜ ∘ 𝐹 ) ∈ MblFn ↔ ∀ 𝑥 ∈ ran (,) ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
17 |
|
imf |
⊢ ℑ : ℂ ⟶ ℝ |
18 |
|
fco |
⊢ ( ( ℑ : ℂ ⟶ ℝ ∧ 𝐹 : 𝐴 ⟶ ℂ ) → ( ℑ ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
19 |
17 18
|
mpan |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( ℑ ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
20 |
19
|
adantr |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ℑ ∘ 𝐹 ) : 𝐴 ⟶ ℝ ) |
21 |
|
ismbf |
⊢ ( ( ℑ ∘ 𝐹 ) : 𝐴 ⟶ ℝ → ( ( ℑ ∘ 𝐹 ) ∈ MblFn ↔ ∀ 𝑥 ∈ ran (,) ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
22 |
20 21
|
syl |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ( ℑ ∘ 𝐹 ) ∈ MblFn ↔ ∀ 𝑥 ∈ ran (,) ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
23 |
16 22
|
anbi12d |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) ↔ ( ∀ 𝑥 ∈ ran (,) ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ∀ 𝑥 ∈ ran (,) ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) |
24 |
|
r19.26 |
⊢ ( ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ↔ ( ∀ 𝑥 ∈ ran (,) ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ∀ 𝑥 ∈ ran (,) ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
25 |
23 24
|
bitr4di |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) ↔ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) |
26 |
|
mblss |
⊢ ( 𝐴 ∈ dom vol → 𝐴 ⊆ ℝ ) |
27 |
|
cnex |
⊢ ℂ ∈ V |
28 |
|
reex |
⊢ ℝ ∈ V |
29 |
|
elpm2r |
⊢ ( ( ( ℂ ∈ V ∧ ℝ ∈ V ) ∧ ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ ℝ ) ) → 𝐹 ∈ ( ℂ ↑pm ℝ ) ) |
30 |
27 28 29
|
mpanl12 |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ ℝ ) → 𝐹 ∈ ( ℂ ↑pm ℝ ) ) |
31 |
26 30
|
sylan2 |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → 𝐹 ∈ ( ℂ ↑pm ℝ ) ) |
32 |
31
|
biantrurd |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ↔ ( 𝐹 ∈ ( ℂ ↑pm ℝ ) ∧ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) ) |
33 |
25 32
|
bitrd |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) ↔ ( 𝐹 ∈ ( ℂ ↑pm ℝ ) ∧ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) ) |
34 |
13 33
|
bitr4id |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐴 ∈ dom vol ) → ( 𝐹 ∈ MblFn ↔ ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) ) ) |
35 |
34
|
ex |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( 𝐴 ∈ dom vol → ( 𝐹 ∈ MblFn ↔ ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) ) ) ) |
36 |
4 12 35
|
pm5.21ndd |
⊢ ( 𝐹 : 𝐴 ⟶ ℂ → ( 𝐹 ∈ MblFn ↔ ( ( ℜ ∘ 𝐹 ) ∈ MblFn ∧ ( ℑ ∘ 𝐹 ) ∈ MblFn ) ) ) |