Step |
Hyp |
Ref |
Expression |
1 |
|
ftc2nc.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ftc2nc.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
ftc2nc.le |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
4 |
|
ftc2nc.c |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) |
5 |
|
ftc2nc.i |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) ∈ 𝐿1 ) |
6 |
|
ftc2nc.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
7 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
8 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
9 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
10 |
7 8 3 9
|
syl3anc |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
11 |
|
fvex |
⊢ ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) ∈ V |
12 |
11
|
fvconst2 |
⊢ ( 𝐵 ∈ ( 𝐴 [,] 𝐵 ) → ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) = ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) ) |
13 |
10 12
|
syl |
⊢ ( 𝜑 → ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) = ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) ) |
14 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
15 |
14
|
subcn |
⊢ − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) |
16 |
15
|
a1i |
⊢ ( 𝜑 → − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
17 |
|
eqid |
⊢ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
18 |
|
ssidd |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 (,) 𝐵 ) ) |
19 |
|
ioossre |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ℝ |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ℝ ) |
21 |
|
cncff |
⊢ ( ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ) |
22 |
4 21
|
syl |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ) |
23 |
|
ioof |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ |
24 |
|
ffun |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → Fun (,) ) |
25 |
23 24
|
ax-mp |
⊢ Fun (,) |
26 |
|
fvelima |
⊢ ( ( Fun (,) ∧ 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ) → ∃ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ( (,) ‘ 𝑥 ) = 𝑠 ) |
27 |
25 26
|
mpan |
⊢ ( 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ( (,) ‘ 𝑥 ) = 𝑠 ) |
28 |
|
1st2nd2 |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → 𝑥 = 〈 ( 1st ‘ 𝑥 ) , ( 2nd ‘ 𝑥 ) 〉 ) |
29 |
28
|
fveq2d |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( (,) ‘ 𝑥 ) = ( (,) ‘ 〈 ( 1st ‘ 𝑥 ) , ( 2nd ‘ 𝑥 ) 〉 ) ) |
30 |
|
df-ov |
⊢ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = ( (,) ‘ 〈 ( 1st ‘ 𝑥 ) , ( 2nd ‘ 𝑥 ) 〉 ) |
31 |
29 30
|
eqtr4di |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( (,) ‘ 𝑥 ) = ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) |
32 |
31
|
eqeq1d |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( ( (,) ‘ 𝑥 ) = 𝑠 ↔ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( (,) ‘ 𝑥 ) = 𝑠 ↔ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 ) ) |
34 |
7 8
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) |
36 |
|
xp1st |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( 1st ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) |
37 |
|
elicc1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 1st ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ↔ ( ( 1st ‘ 𝑥 ) ∈ ℝ* ∧ 𝐴 ≤ ( 1st ‘ 𝑥 ) ∧ ( 1st ‘ 𝑥 ) ≤ 𝐵 ) ) ) |
38 |
7 8 37
|
syl2anc |
⊢ ( 𝜑 → ( ( 1st ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ↔ ( ( 1st ‘ 𝑥 ) ∈ ℝ* ∧ 𝐴 ≤ ( 1st ‘ 𝑥 ) ∧ ( 1st ‘ 𝑥 ) ≤ 𝐵 ) ) ) |
39 |
38
|
biimpa |
⊢ ( ( 𝜑 ∧ ( 1st ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 1st ‘ 𝑥 ) ∈ ℝ* ∧ 𝐴 ≤ ( 1st ‘ 𝑥 ) ∧ ( 1st ‘ 𝑥 ) ≤ 𝐵 ) ) |
40 |
39
|
simp2d |
⊢ ( ( 𝜑 ∧ ( 1st ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ ( 1st ‘ 𝑥 ) ) |
41 |
36 40
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → 𝐴 ≤ ( 1st ‘ 𝑥 ) ) |
42 |
|
xp2nd |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( 2nd ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) |
43 |
|
iccleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 2nd ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 2nd ‘ 𝑥 ) ≤ 𝐵 ) |
44 |
43
|
3expa |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 2nd ‘ 𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 2nd ‘ 𝑥 ) ≤ 𝐵 ) |
45 |
34 42 44
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 2nd ‘ 𝑥 ) ≤ 𝐵 ) |
46 |
|
ioossioo |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ ( 1st ‘ 𝑥 ) ∧ ( 2nd ‘ 𝑥 ) ≤ 𝐵 ) ) → ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ⊆ ( 𝐴 (,) 𝐵 ) ) |
47 |
35 41 45 46
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ⊆ ( 𝐴 (,) 𝐵 ) ) |
48 |
47
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) → 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) |
49 |
22
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ ℂ ) |
50 |
49
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ ℂ ) |
51 |
48 50
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ ℂ ) |
52 |
|
ioombl |
⊢ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ∈ dom vol |
53 |
52
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ∈ dom vol ) |
54 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V ) |
55 |
22
|
feqmptd |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) = ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) |
56 |
55 5
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 ) |
57 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 ) |
58 |
47 53 54 57
|
iblss |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 ) |
59 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
60 |
|
ssid |
⊢ ℂ ⊆ ℂ |
61 |
|
cncfss |
⊢ ( ( ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ℂ –cn→ ℝ ) ⊆ ( ℂ –cn→ ℂ ) ) |
62 |
59 60 61
|
mp2an |
⊢ ( ℂ –cn→ ℝ ) ⊆ ( ℂ –cn→ ℂ ) |
63 |
|
abscncf |
⊢ abs ∈ ( ℂ –cn→ ℝ ) |
64 |
62 63
|
sselii |
⊢ abs ∈ ( ℂ –cn→ ℂ ) |
65 |
64
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → abs ∈ ( ℂ –cn→ ℂ ) ) |
66 |
55
|
reseq1d |
⊢ ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) = ( ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) ) |
67 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) = ( ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) ) |
68 |
47
|
resmptd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) = ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) |
69 |
67 68
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) = ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) |
70 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) |
71 |
|
rescncf |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ⊆ ( 𝐴 (,) 𝐵 ) → ( ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) ) |
72 |
47 70 71
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
73 |
69 72
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
74 |
65 73
|
cncfmpt1f |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
75 |
|
cnmbf |
⊢ ( ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ∈ dom vol ∧ ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn ) |
76 |
52 74 75
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn ) |
77 |
51 58
|
itgcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ ) |
78 |
77
|
cjcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ℂ ) |
79 |
|
ioossre |
⊢ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ⊆ ℝ |
80 |
79 59
|
sstri |
⊢ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ⊆ ℂ |
81 |
|
cncfmptc |
⊢ ( ( ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ℂ ∧ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
82 |
80 60 81
|
mp3an23 |
⊢ ( ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ℂ → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
83 |
78 82
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
84 |
|
nfcv |
⊢ Ⅎ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) |
85 |
|
nfcsb1v |
⊢ Ⅎ 𝑡 ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) |
86 |
|
csbeq1a |
⊢ ( 𝑡 = 𝑠 → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) = ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) |
87 |
84 85 86
|
cbvmpt |
⊢ ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) = ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) |
88 |
87 73
|
eqeltrrid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
89 |
83 88
|
mulcncf |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) |
90 |
|
cnmbf |
⊢ ( ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ∈ dom vol ∧ ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) –cn→ ℂ ) ) → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn ) |
91 |
52 89 90
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · ⦋ 𝑠 / 𝑡 ⦌ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn ) |
92 |
51 58 76 91
|
itgabsnc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( abs ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) d 𝑡 ) |
93 |
51
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) → ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ ℝ ) |
94 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V ) |
95 |
94 58 76
|
iblabsnc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ 𝐿1 ) |
96 |
51
|
absge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ) → 0 ≤ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) |
97 |
93 95 96
|
itgposval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) d 𝑡 = ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) |
98 |
92 97
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( abs ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) |
99 |
|
itgeq1 |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
100 |
99
|
fveq2d |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ( abs ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) = ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) |
101 |
|
eleq2 |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ↔ 𝑡 ∈ 𝑠 ) ) |
102 |
101
|
ifbid |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → if ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) = if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) |
103 |
102
|
mpteq2dv |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) = ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) |
104 |
103
|
fveq2d |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) = ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) |
105 |
100 104
|
breq12d |
⊢ ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ( ( abs ‘ ∫ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ↔ ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) ) |
106 |
98 105
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ( 1st ‘ 𝑥 ) (,) ( 2nd ‘ 𝑥 ) ) = 𝑠 → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) ) |
107 |
33 106
|
sylbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( (,) ‘ 𝑥 ) = 𝑠 → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) ) |
108 |
107
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ( (,) ‘ 𝑥 ) = 𝑠 → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) ) |
109 |
27 108
|
syl5 |
⊢ ( 𝜑 → ( 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) ) |
110 |
109
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ 𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) |
111 |
17 1 2 3 18 20 5 22 110
|
ftc1anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
112 |
|
cncff |
⊢ ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
113 |
6 112
|
syl |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
114 |
113
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
115 |
114 6
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
116 |
14 16 111 115
|
cncfmpt2f |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
117 |
59
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
118 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
119 |
1 2 118
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
120 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝑥 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V ) |
121 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ ) |
122 |
121
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ* ) |
123 |
|
elicc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵 ) ) ) |
124 |
1 2 123
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵 ) ) ) |
125 |
124
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵 ) ) |
126 |
125
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ≤ 𝐵 ) |
127 |
|
iooss2 |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝑥 ≤ 𝐵 ) → ( 𝐴 (,) 𝑥 ) ⊆ ( 𝐴 (,) 𝐵 ) ) |
128 |
122 126 127
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 (,) 𝑥 ) ⊆ ( 𝐴 (,) 𝐵 ) ) |
129 |
|
ioombl |
⊢ ( 𝐴 (,) 𝑥 ) ∈ dom vol |
130 |
129
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 (,) 𝑥 ) ∈ dom vol ) |
131 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V ) |
132 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 ) |
133 |
128 130 131 132
|
iblss |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑡 ∈ ( 𝐴 (,) 𝑥 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 ) |
134 |
120 133
|
itgcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ ) |
135 |
113
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
136 |
134 135
|
subcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ∈ ℂ ) |
137 |
14
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
138 |
|
iccntr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
139 |
1 2 138
|
syl2anc |
⊢ ( 𝜑 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) |
140 |
117 119 136 137 14 139
|
dvmptntr |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
141 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
142 |
141
|
a1i |
⊢ ( 𝜑 → ℝ ∈ { ℝ , ℂ } ) |
143 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
144 |
143
|
sseli |
⊢ ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
145 |
144 134
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ ) |
146 |
22
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ∈ ℂ ) |
147 |
17 1 2 3 4 5
|
ftc1cnnc |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) = ( ℝ D 𝐹 ) ) |
148 |
117 119 134 137 14 139
|
dvmptntr |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ) |
149 |
22
|
feqmptd |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
150 |
147 148 149
|
3eqtr3d |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
151 |
144 135
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
152 |
114
|
oveq2d |
⊢ ( 𝜑 → ( ℝ D 𝐹 ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) ) |
153 |
117 119 135 137 14 139
|
dvmptntr |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) ) |
154 |
152 149 153
|
3eqtr3rd |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
155 |
142 145 146 150 151 146 154
|
dvmptsub |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) − ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) ) |
156 |
146
|
subidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) − ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = 0 ) |
157 |
156
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) − ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 0 ) ) |
158 |
140 155 157
|
3eqtrd |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 0 ) ) |
159 |
|
fconstmpt |
⊢ ( ( 𝐴 (,) 𝐵 ) × { 0 } ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 0 ) |
160 |
158 159
|
eqtr4di |
⊢ ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ) = ( ( 𝐴 (,) 𝐵 ) × { 0 } ) ) |
161 |
1 2 116 160
|
dveq0 |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) = ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) } ) ) |
162 |
161
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐵 ) = ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) ) |
163 |
|
oveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐴 (,) 𝑥 ) = ( 𝐴 (,) 𝐵 ) ) |
164 |
|
itgeq1 |
⊢ ( ( 𝐴 (,) 𝑥 ) = ( 𝐴 (,) 𝐵 ) → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
165 |
163 164
|
syl |
⊢ ( 𝑥 = 𝐵 → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
166 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐵 ) ) |
167 |
165 166
|
oveq12d |
⊢ ( 𝑥 = 𝐵 → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ) |
168 |
|
eqid |
⊢ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) |
169 |
|
ovex |
⊢ ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ∈ V |
170 |
167 168 169
|
fvmpt |
⊢ ( 𝐵 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐵 ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ) |
171 |
10 170
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐵 ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ) |
172 |
162 171
|
eqtr3d |
⊢ ( 𝜑 → ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ) |
173 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
174 |
7 8 3 173
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
175 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐴 (,) 𝑥 ) = ( 𝐴 (,) 𝐴 ) ) |
176 |
|
iooid |
⊢ ( 𝐴 (,) 𝐴 ) = ∅ |
177 |
175 176
|
eqtrdi |
⊢ ( 𝑥 = 𝐴 → ( 𝐴 (,) 𝑥 ) = ∅ ) |
178 |
|
itgeq1 |
⊢ ( ( 𝐴 (,) 𝑥 ) = ∅ → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ∅ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
179 |
177 178
|
syl |
⊢ ( 𝑥 = 𝐴 → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ∅ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
180 |
|
itg0 |
⊢ ∫ ∅ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = 0 |
181 |
179 180
|
eqtrdi |
⊢ ( 𝑥 = 𝐴 → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = 0 ) |
182 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐴 ) ) |
183 |
181 182
|
oveq12d |
⊢ ( 𝑥 = 𝐴 → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) = ( 0 − ( 𝐹 ‘ 𝐴 ) ) ) |
184 |
|
df-neg |
⊢ - ( 𝐹 ‘ 𝐴 ) = ( 0 − ( 𝐹 ‘ 𝐴 ) ) |
185 |
183 184
|
eqtr4di |
⊢ ( 𝑥 = 𝐴 → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) = - ( 𝐹 ‘ 𝐴 ) ) |
186 |
|
negex |
⊢ - ( 𝐹 ‘ 𝐴 ) ∈ V |
187 |
185 168 186
|
fvmpt |
⊢ ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) = - ( 𝐹 ‘ 𝐴 ) ) |
188 |
174 187
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝑥 ) ) ) ‘ 𝐴 ) = - ( 𝐹 ‘ 𝐴 ) ) |
189 |
13 172 188
|
3eqtr3d |
⊢ ( 𝜑 → ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) = - ( 𝐹 ‘ 𝐴 ) ) |
190 |
189
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐵 ) + ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ) = ( ( 𝐹 ‘ 𝐵 ) + - ( 𝐹 ‘ 𝐴 ) ) ) |
191 |
113 10
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ℂ ) |
192 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V ) |
193 |
192 56
|
itgcl |
⊢ ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ ) |
194 |
191 193
|
pncan3d |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐵 ) + ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹 ‘ 𝐵 ) ) ) = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) |
195 |
113 174
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ℂ ) |
196 |
191 195
|
negsubd |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐵 ) + - ( 𝐹 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) |
197 |
190 194 196
|
3eqtr3d |
⊢ ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ( ( 𝐹 ‘ 𝐵 ) − ( 𝐹 ‘ 𝐴 ) ) ) |