Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) |
2 |
|
cncff |
⊢ ( 𝐺 ∈ ( 𝐵 –cn→ ℂ ) → 𝐺 : 𝐵 ⟶ ℂ ) |
3 |
1 2
|
syl |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → 𝐺 : 𝐵 ⟶ ℂ ) |
4 |
|
simp2 |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
5 |
|
fco |
⊢ ( ( 𝐺 : 𝐵 ⟶ ℂ ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ( 𝐺 ∘ 𝐹 ) : 𝐴 ⟶ ℂ ) |
6 |
3 4 5
|
syl2anc |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ( 𝐺 ∘ 𝐹 ) : 𝐴 ⟶ ℂ ) |
7 |
4
|
fdmd |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → dom 𝐹 = 𝐴 ) |
8 |
|
mbfdm |
⊢ ( 𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol ) |
9 |
8
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → dom 𝐹 ∈ dom vol ) |
10 |
7 9
|
eqeltrrd |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → 𝐴 ∈ dom vol ) |
11 |
|
mblss |
⊢ ( 𝐴 ∈ dom vol → 𝐴 ⊆ ℝ ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → 𝐴 ⊆ ℝ ) |
13 |
|
cnex |
⊢ ℂ ∈ V |
14 |
|
reex |
⊢ ℝ ∈ V |
15 |
|
elpm2r |
⊢ ( ( ( ℂ ∈ V ∧ ℝ ∈ V ) ∧ ( ( 𝐺 ∘ 𝐹 ) : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ ℝ ) ) → ( 𝐺 ∘ 𝐹 ) ∈ ( ℂ ↑pm ℝ ) ) |
16 |
13 14 15
|
mpanl12 |
⊢ ( ( ( 𝐺 ∘ 𝐹 ) : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ ℝ ) → ( 𝐺 ∘ 𝐹 ) ∈ ( ℂ ↑pm ℝ ) ) |
17 |
6 12 16
|
syl2anc |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ( 𝐺 ∘ 𝐹 ) ∈ ( ℂ ↑pm ℝ ) ) |
18 |
|
coeq1 |
⊢ ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( 𝑔 ∘ 𝐹 ) = ( ( ℜ ∘ 𝐺 ) ∘ 𝐹 ) ) |
19 |
|
coass |
⊢ ( ( ℜ ∘ 𝐺 ) ∘ 𝐹 ) = ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) |
20 |
18 19
|
eqtrdi |
⊢ ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( 𝑔 ∘ 𝐹 ) = ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) ) |
21 |
20
|
cnveqd |
⊢ ( 𝑔 = ( ℜ ∘ 𝐺 ) → ◡ ( 𝑔 ∘ 𝐹 ) = ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) ) |
22 |
21
|
imaeq1d |
⊢ ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) = ( ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ) |
23 |
22
|
eleq1d |
⊢ ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ↔ ( ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) ) |
24 |
|
cnvco |
⊢ ◡ ( 𝑔 ∘ 𝐹 ) = ( ◡ 𝐹 ∘ ◡ 𝑔 ) |
25 |
24
|
imaeq1i |
⊢ ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) = ( ( ◡ 𝐹 ∘ ◡ 𝑔 ) “ 𝑥 ) |
26 |
|
imaco |
⊢ ( ( ◡ 𝐹 ∘ ◡ 𝑔 ) “ 𝑥 ) = ( ◡ 𝐹 “ ( ◡ 𝑔 “ 𝑥 ) ) |
27 |
25 26
|
eqtri |
⊢ ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) = ( ◡ 𝐹 “ ( ◡ 𝑔 “ 𝑥 ) ) |
28 |
|
simplll |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝐹 ∈ MblFn ) |
29 |
|
simpllr |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
30 |
|
cncfrss |
⊢ ( 𝑔 ∈ ( 𝐵 –cn→ ℝ ) → 𝐵 ⊆ ℂ ) |
31 |
30
|
adantl |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝐵 ⊆ ℂ ) |
32 |
|
simpr |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) |
33 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
34 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
35 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) |
36 |
34
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
37 |
34 35 36
|
cncfcn |
⊢ ( ( 𝐵 ⊆ ℂ ∧ ℝ ⊆ ℂ ) → ( 𝐵 –cn→ ℝ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) ) |
38 |
31 33 37
|
sylancl |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → ( 𝐵 –cn→ ℝ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) ) |
39 |
32 38
|
eleqtrd |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝑔 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) ) |
40 |
|
retopbas |
⊢ ran (,) ∈ TopBases |
41 |
|
bastg |
⊢ ( ran (,) ∈ TopBases → ran (,) ⊆ ( topGen ‘ ran (,) ) ) |
42 |
40 41
|
ax-mp |
⊢ ran (,) ⊆ ( topGen ‘ ran (,) ) |
43 |
|
simplr |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝑥 ∈ ran (,) ) |
44 |
42 43
|
sselid |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → 𝑥 ∈ ( topGen ‘ ran (,) ) ) |
45 |
|
cnima |
⊢ ( ( 𝑔 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ ( topGen ‘ ran (,) ) ) → ( ◡ 𝑔 “ 𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) ) |
46 |
39 44 45
|
syl2anc |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → ( ◡ 𝑔 “ 𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) ) |
47 |
34 35
|
mbfimaopn2 |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ⊆ ℂ ) ∧ ( ◡ 𝑔 “ 𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) ) → ( ◡ 𝐹 “ ( ◡ 𝑔 “ 𝑥 ) ) ∈ dom vol ) |
48 |
28 29 31 46 47
|
syl31anc |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → ( ◡ 𝐹 “ ( ◡ 𝑔 “ 𝑥 ) ) ∈ dom vol ) |
49 |
27 48
|
eqeltrid |
⊢ ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ) → ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) |
50 |
49
|
ralrimiva |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) ∧ 𝑥 ∈ ran (,) ) → ∀ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) |
51 |
50
|
3adantl3 |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ∀ 𝑔 ∈ ( 𝐵 –cn→ ℝ ) ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) |
52 |
|
recncf |
⊢ ℜ ∈ ( ℂ –cn→ ℝ ) |
53 |
52
|
a1i |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ℜ ∈ ( ℂ –cn→ ℝ ) ) |
54 |
1 53
|
cncfco |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ( ℜ ∘ 𝐺 ) ∈ ( 𝐵 –cn→ ℝ ) ) |
55 |
54
|
adantr |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ℜ ∘ 𝐺 ) ∈ ( 𝐵 –cn→ ℝ ) ) |
56 |
23 51 55
|
rspcdva |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) |
57 |
|
coeq1 |
⊢ ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( 𝑔 ∘ 𝐹 ) = ( ( ℑ ∘ 𝐺 ) ∘ 𝐹 ) ) |
58 |
|
coass |
⊢ ( ( ℑ ∘ 𝐺 ) ∘ 𝐹 ) = ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) |
59 |
57 58
|
eqtrdi |
⊢ ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( 𝑔 ∘ 𝐹 ) = ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) ) |
60 |
59
|
cnveqd |
⊢ ( 𝑔 = ( ℑ ∘ 𝐺 ) → ◡ ( 𝑔 ∘ 𝐹 ) = ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) ) |
61 |
60
|
imaeq1d |
⊢ ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) = ( ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ) |
62 |
61
|
eleq1d |
⊢ ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( ( ◡ ( 𝑔 ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ↔ ( ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) ) |
63 |
|
imcncf |
⊢ ℑ ∈ ( ℂ –cn→ ℝ ) |
64 |
63
|
a1i |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ℑ ∈ ( ℂ –cn→ ℝ ) ) |
65 |
1 64
|
cncfco |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ( ℑ ∘ 𝐺 ) ∈ ( 𝐵 –cn→ ℝ ) ) |
66 |
65
|
adantr |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ℑ ∘ 𝐺 ) ∈ ( 𝐵 –cn→ ℝ ) ) |
67 |
62 51 66
|
rspcdva |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) |
68 |
56 67
|
jca |
⊢ ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ( ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) ) |
69 |
68
|
ralrimiva |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) ) |
70 |
|
ismbf1 |
⊢ ( ( 𝐺 ∘ 𝐹 ) ∈ MblFn ↔ ( ( 𝐺 ∘ 𝐹 ) ∈ ( ℂ ↑pm ℝ ) ∧ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ ( 𝐺 ∘ 𝐹 ) ) “ 𝑥 ) ∈ dom vol ) ) ) |
71 |
17 69 70
|
sylanbrc |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 ∈ ( 𝐵 –cn→ ℂ ) ) → ( 𝐺 ∘ 𝐹 ) ∈ MblFn ) |