Step |
Hyp |
Ref |
Expression |
1 |
|
ftc2re.e |
⊢ 𝐸 = ( 𝐶 (,) 𝐷 ) |
2 |
|
ftc2re.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐸 ) |
3 |
|
ftc2re.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝐸 ) |
4 |
|
ftc2re.le |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
5 |
|
ftc2re.f |
⊢ ( 𝜑 → 𝐹 : 𝐸 ⟶ ℂ ) |
6 |
|
ftc2re.1 |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) ∈ ( 𝐸 –cn→ ℂ ) ) |
7 |
|
ioossre |
⊢ ( 𝐶 (,) 𝐷 ) ⊆ ℝ |
8 |
1 7
|
eqsstri |
⊢ 𝐸 ⊆ ℝ |
9 |
8
|
a1i |
⊢ ( 𝜑 → 𝐸 ⊆ ℝ ) |
10 |
9 2
|
sseldd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
11 |
9 3
|
sseldd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
12 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
13 |
12
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
14 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
15 |
10 11 14
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
16 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
17 |
16
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
18 |
16 17
|
dvres |
⊢ ( ( ( ℝ ⊆ ℂ ∧ 𝐹 : 𝐸 ⟶ ℂ ) ∧ ( 𝐸 ⊆ ℝ ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) ) → ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) ) ) |
19 |
13 5 9 15 18
|
syl22anc |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) ) ) |
20 |
|
iccntr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
21 |
10 11 20
|
syl2anc |
⊢ ( 𝜑 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
22 |
21
|
reseq2d |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) |
23 |
19 22
|
eqtrd |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) |
24 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
25 |
24
|
a1i |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
26 |
1 2 3
|
fct2relem |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ 𝐸 ) |
27 |
25 26
|
sstrd |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ 𝐸 ) |
28 |
|
rescncf |
⊢ ( ( 𝐴 (,) 𝐵 ) ⊆ 𝐸 → ( ( ℝ D 𝐹 ) ∈ ( 𝐸 –cn→ ℂ ) → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) ) |
29 |
27 6 28
|
sylc |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) |
30 |
23 29
|
eqeltrd |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) |
31 |
|
ioombl |
⊢ ( 𝐴 (,) 𝐵 ) ∈ dom vol |
32 |
31
|
a1i |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ∈ dom vol ) |
33 |
|
cnmbf |
⊢ ( ( ( 𝐴 (,) 𝐵 ) ∈ dom vol ∧ ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ MblFn ) |
34 |
32 29 33
|
syl2anc |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ MblFn ) |
35 |
|
dmres |
⊢ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) = ( ( 𝐴 (,) 𝐵 ) ∩ dom ( ℝ D 𝐹 ) ) |
36 |
35
|
fveq2i |
⊢ ( vol ‘ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) = ( vol ‘ ( ( 𝐴 (,) 𝐵 ) ∩ dom ( ℝ D 𝐹 ) ) ) |
37 |
|
cncff |
⊢ ( ( ℝ D 𝐹 ) ∈ ( 𝐸 –cn→ ℂ ) → ( ℝ D 𝐹 ) : 𝐸 ⟶ ℂ ) |
38 |
6 37
|
syl |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) : 𝐸 ⟶ ℂ ) |
39 |
38
|
fdmd |
⊢ ( 𝜑 → dom ( ℝ D 𝐹 ) = 𝐸 ) |
40 |
39
|
ineq2d |
⊢ ( 𝜑 → ( ( 𝐴 (,) 𝐵 ) ∩ dom ( ℝ D 𝐹 ) ) = ( ( 𝐴 (,) 𝐵 ) ∩ 𝐸 ) ) |
41 |
|
df-ss |
⊢ ( ( 𝐴 (,) 𝐵 ) ⊆ 𝐸 ↔ ( ( 𝐴 (,) 𝐵 ) ∩ 𝐸 ) = ( 𝐴 (,) 𝐵 ) ) |
42 |
27 41
|
sylib |
⊢ ( 𝜑 → ( ( 𝐴 (,) 𝐵 ) ∩ 𝐸 ) = ( 𝐴 (,) 𝐵 ) ) |
43 |
40 42
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐴 (,) 𝐵 ) ∩ dom ( ℝ D 𝐹 ) ) = ( 𝐴 (,) 𝐵 ) ) |
44 |
43
|
fveq2d |
⊢ ( 𝜑 → ( vol ‘ ( ( 𝐴 (,) 𝐵 ) ∩ dom ( ℝ D 𝐹 ) ) ) = ( vol ‘ ( 𝐴 (,) 𝐵 ) ) ) |
45 |
|
volioo |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐵 − 𝐴 ) ) |
46 |
10 11 4 45
|
syl3anc |
⊢ ( 𝜑 → ( vol ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐵 − 𝐴 ) ) |
47 |
11 10
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
48 |
46 47
|
eqeltrd |
⊢ ( 𝜑 → ( vol ‘ ( 𝐴 (,) 𝐵 ) ) ∈ ℝ ) |
49 |
44 48
|
eqeltrd |
⊢ ( 𝜑 → ( vol ‘ ( ( 𝐴 (,) 𝐵 ) ∩ dom ( ℝ D 𝐹 ) ) ) ∈ ℝ ) |
50 |
36 49
|
eqeltrid |
⊢ ( 𝜑 → ( vol ‘ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) ∈ ℝ ) |
51 |
|
rescncf |
⊢ ( ( 𝐴 [,] 𝐵 ) ⊆ 𝐸 → ( ( ℝ D 𝐹 ) ∈ ( 𝐸 –cn→ ℂ ) → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) ) |
52 |
26 51
|
syl |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ∈ ( 𝐸 –cn→ ℂ ) → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) ) |
53 |
6 52
|
mpd |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
54 |
|
cniccbdd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) |
55 |
10 11 53 54
|
syl3anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) |
56 |
35 43
|
syl5eq |
⊢ ( 𝜑 → dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
57 |
56 25
|
eqsstrd |
⊢ ( 𝜑 → dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
58 |
|
ssralv |
⊢ ( dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) → ( ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
59 |
57 58
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
61 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
62 |
61
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) |
63 |
|
fvres |
⊢ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) → ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) |
64 |
62 63
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) |
65 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) |
66 |
56
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
67 |
65 66
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ) |
68 |
|
fvres |
⊢ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) → ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) = ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) |
69 |
67 68
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) = ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) |
70 |
64 69
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) |
71 |
70
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) = ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ) |
72 |
71
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → ( ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ↔ ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
73 |
72
|
biimpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) → ( ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
74 |
73
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
75 |
60 74
|
syld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
76 |
75
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
77 |
55 76
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) |
78 |
|
bddibl |
⊢ ( ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ MblFn ∧ ( vol ‘ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ) ∈ ℝ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ dom ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ( abs ‘ ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑦 ) ) ≤ 𝑥 ) → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐿1 ) |
79 |
34 50 77 78
|
syl3anc |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐿1 ) |
80 |
23 79
|
eqeltrd |
⊢ ( 𝜑 → ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ∈ 𝐿1 ) |
81 |
|
dvcn |
⊢ ( ( ( ℝ ⊆ ℂ ∧ 𝐹 : 𝐸 ⟶ ℂ ∧ 𝐸 ⊆ ℝ ) ∧ dom ( ℝ D 𝐹 ) = 𝐸 ) → 𝐹 ∈ ( 𝐸 –cn→ ℂ ) ) |
82 |
13 5 9 39 81
|
syl31anc |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐸 –cn→ ℂ ) ) |
83 |
|
rescncf |
⊢ ( ( 𝐴 [,] 𝐵 ) ⊆ 𝐸 → ( 𝐹 ∈ ( 𝐸 –cn→ ℂ ) → ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) ) |
84 |
26 83
|
syl |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐸 –cn→ ℂ ) → ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) ) |
85 |
82 84
|
mpd |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
86 |
10 11 4 30 80 85
|
ftc2 |
⊢ ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) d 𝑡 = ( ( ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) ) |
87 |
23
|
fveq1d |
⊢ ( 𝜑 → ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) = ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑡 ) ) |
88 |
|
fvres |
⊢ ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) → ( ( ( ℝ D 𝐹 ) ↾ ( 𝐴 (,) 𝐵 ) ) ‘ 𝑡 ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) |
89 |
87 88
|
sylan9eq |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) |
90 |
89
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) |
91 |
|
itgeq2 |
⊢ ( ∀ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) = ( ( ℝ D 𝐹 ) ‘ 𝑡 ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) d 𝑡 = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
92 |
90 91
|
syl |
⊢ ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ) ‘ 𝑡 ) d 𝑡 = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
93 |
10
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
94 |
11
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
95 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
96 |
93 94 4 95
|
syl3anc |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
97 |
96
|
fvresd |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) ) |
98 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
99 |
93 94 4 98
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
100 |
99
|
fvresd |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) |
101 |
97 100
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐵 ) − ( ( 𝐹 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝐴 ) ) = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) |
102 |
86 92 101
|
3eqtr3d |
⊢ ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) |