Step | Hyp | Ref
| Expression |
1 | | ftc2nc.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | 1 | rexrd 9968 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
3 | | ftc2nc.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
4 | 3 | rexrd 9968 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
5 | | ftc2nc.le |
. . . . . 6
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | | ubicc2 12160 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
7 | 2, 4, 5, 6 | syl3anc 1318 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
8 | | fvex 6113 |
. . . . . 6
⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) ∈ V |
9 | 8 | fvconst2 6374 |
. . . . 5
⊢ (𝐵 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)) |
10 | 7, 9 | syl 17 |
. . . 4
⊢ (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)) |
11 | | eqid 2610 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
12 | 11 | subcn 22477 |
. . . . . . . . 9
⊢ −
∈ (((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
13 | 12 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → − ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
14 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) |
15 | | ssid 3587 |
. . . . . . . . . 10
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵) |
16 | 15 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) |
17 | | ioossre 12106 |
. . . . . . . . . 10
⊢ (𝐴(,)𝐵) ⊆ ℝ |
18 | 17 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
19 | | ftc2nc.i |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D 𝐹) ∈
𝐿1) |
20 | | ftc2nc.c |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
21 | | cncff 22504 |
. . . . . . . . . 10
⊢ ((ℝ
D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ) |
22 | 20, 21 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ) |
23 | | ioof 12142 |
. . . . . . . . . . . . 13
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
24 | | ffun 5961 |
. . . . . . . . . . . . 13
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → Fun (,)) |
25 | 23, 24 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
(,) |
26 | | fvelima 6158 |
. . . . . . . . . . . 12
⊢ ((Fun (,)
∧ 𝑠 ∈ ((,) “
((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠) |
27 | 25, 26 | mpan 702 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠) |
28 | | 1st2nd2 7096 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
29 | 28 | fveq2d 6107 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘〈(1st
‘𝑥), (2nd
‘𝑥)〉)) |
30 | | df-ov 6552 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘〈(1st
‘𝑥), (2nd
‘𝑥)〉) |
31 | 29, 30 | syl6eqr 2662 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥))) |
32 | 31 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠)) |
33 | 32 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠)) |
34 | 2, 4 | jca 553 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈
ℝ*)) |
35 | 34 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈
ℝ*)) |
36 | | xp1st 7089 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st ‘𝑥) ∈ (𝐴[,]𝐵)) |
37 | | elicc1 12090 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((1st ‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵))) |
38 | 2, 4, 37 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1st
‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵))) |
39 | 38 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (1st
‘𝑥) ∈ (𝐴[,]𝐵)) → ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵)) |
40 | 39 | simp2d 1067 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (1st
‘𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st ‘𝑥)) |
41 | 36, 40 | sylan2 490 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st ‘𝑥)) |
42 | | xp2nd 7090 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) |
43 | | iccleub 12100 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵) |
44 | 43 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵) |
45 | 34, 42, 44 | syl2an 493 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd ‘𝑥) ≤ 𝐵) |
46 | | ioossioo 12136 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐴 ≤ (1st ‘𝑥) ∧ (2nd
‘𝑥) ≤ 𝐵)) → ((1st
‘𝑥)(,)(2nd
‘𝑥)) ⊆ (𝐴(,)𝐵)) |
47 | 35, 41, 45, 46 | syl12anc 1316 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵)) |
48 | 47 | sselda 3568 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 𝑡 ∈ (𝐴(,)𝐵)) |
49 | 22 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
50 | 49 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
51 | 48, 50 | syldan 486 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
52 | | ioombl 23140 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol |
53 | 52 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom
vol) |
54 | | fvex 6113 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℝ
D 𝐹)‘𝑡) ∈ V |
55 | 54 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
56 | 22 | feqmptd 6159 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡))) |
57 | 56, 19 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
58 | 57 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
59 | 47, 53, 55, 58 | iblss 23377 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
60 | | ax-resscn 9872 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℝ
⊆ ℂ |
61 | | ssid 3587 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℂ
⊆ ℂ |
62 | | cncfss 22510 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)) |
63 | 60, 61, 62 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(ℂ–cn→ℝ)
⊆ (ℂ–cn→ℂ) |
64 | | abscncf 22512 |
. . . . . . . . . . . . . . . . . . . 20
⊢ abs
∈ (ℂ–cn→ℝ) |
65 | 63, 64 | sselii 3565 |
. . . . . . . . . . . . . . . . . . 19
⊢ abs
∈ (ℂ–cn→ℂ) |
66 | 65 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ)) |
67 | 56 | reseq1d 5316 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((1st
‘𝑥)(,)(2nd
‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥)))) |
68 | 67 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥)))) |
69 | 47 | resmptd 5371 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((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))) |
70 | 68, 69 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡))) |
71 | 20 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
72 | | rescncf 22508 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((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))) |
73 | 47, 71, 72 | sylc 63 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) |
74 | 70, 73 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
75 | 66, 74 | cncfmpt1f 22524 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) |
76 | | cnmbf 23232 |
. . . . . . . . . . . . . . . . 17
⊢
((((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) |
77 | 52, 75, 76 | sylancr 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
MblFn) |
78 | 51, 59 | itgcl 23356 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
79 | 78 | cjcld 13784 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) →
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ) |
80 | | ioossre 12106 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℝ |
81 | 80, 60 | sstri 3577 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℂ |
82 | | cncfmptc 22522 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ ∧ ((1st
‘𝑥)(,)(2nd
‘𝑥)) ⊆ ℂ
∧ ℂ ⊆ ℂ) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
83 | 81, 61, 82 | mp3an23 1408 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
84 | 79, 83 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
85 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠((ℝ D 𝐹)‘𝑡) |
86 | | nfcsb1v 3515 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡) |
87 | | csbeq1a 3508 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) |
88 | 85, 86, 87 | cbvmpt 4677 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)) ↦
((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) |
89 | 88, 74 | syl5eqelr 2693 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
90 | 84, 89 | mulcncf 23023 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
91 | | cnmbf 23232 |
. . . . . . . . . . . . . . . . 17
⊢
((((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol ∧ (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋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) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ MblFn) |
92 | 52, 90, 91 | sylancr 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ MblFn) |
93 | 51, 59, 77, 92 | itgabsnc 32649 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st ‘𝑥)(,)(2nd ‘𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡) |
94 | 51 | abscld 14023 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → (abs‘((ℝ
D 𝐹)‘𝑡)) ∈
ℝ) |
95 | 54 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
96 | 95, 59, 77 | iblabsnc 32644 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
𝐿1) |
97 | 51 | absge0d 14031 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 0 ≤
(abs‘((ℝ D 𝐹)‘𝑡))) |
98 | 94, 96, 97 | itgposval 23368 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st
‘𝑥)(,)(2nd
‘𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡 = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
99 | 93, 98 | breqtrd 4609 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
100 | | itgeq1 23345 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) |
101 | 100 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)) |
102 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↔ 𝑡 ∈ 𝑠)) |
103 | 102 | ifbid 4058 |
. . . . . . . . . . . . . . . . 17
⊢
(((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)) |
104 | 103 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . 16
⊢
(((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))) |
105 | 104 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢
(((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)))) |
106 | 101, 105 | breq12d 4596 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ((abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0))) ↔ (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
107 | 99, 106 | syl5ibcom 234 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
108 | 33, 107 | sylbid 229 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
109 | 108 | rexlimdva 3013 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
110 | 27, 109 | syl5 33 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
111 | 110 | ralrimiv 2948 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
112 | 14, 1, 3, 5, 16, 18, 19, 22, 111 | ftc1anc 32663 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
113 | | ftc2nc.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
114 | | cncff 22504 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
115 | 113, 114 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
116 | 115 | feqmptd 6159 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) |
117 | 116, 113 | eqeltrrd 2689 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
118 | 11, 13, 112, 117 | cncfmpt2f 22525 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
119 | 60 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
120 | | iccssre 12126 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
121 | 1, 3, 120 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
122 | 54 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
123 | 3 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
124 | 123 | rexrd 9968 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈
ℝ*) |
125 | | elicc2 12109 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
126 | 1, 3, 125 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
127 | 126 | biimpa 500 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
128 | 127 | simp3d 1068 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) |
129 | | iooss2 12082 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℝ*
∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
130 | 124, 128,
129 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
131 | | ioombl 23140 |
. . . . . . . . . . . . . 14
⊢ (𝐴(,)𝑥) ∈ dom vol |
132 | 131 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol) |
133 | 54 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
134 | 57 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
135 | 130, 132,
133, 134 | iblss 23377 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
136 | 122, 135 | itgcl 23356 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
137 | 115 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
138 | 136, 137 | subcld 10271 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) ∈ ℂ) |
139 | 11 | tgioo2 22414 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
140 | | iccntr 22432 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
141 | 1, 3, 140 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
142 | 119, 121,
138, 139, 11, 141 | dvmptntr 23540 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))))) |
143 | | reelprrecn 9907 |
. . . . . . . . . . 11
⊢ ℝ
∈ {ℝ, ℂ} |
144 | 143 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
145 | | ioossicc 12130 |
. . . . . . . . . . . 12
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
146 | 145 | sseli 3564 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
147 | 146, 136 | sylan2 490 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
148 | 22 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ) |
149 | 14, 1, 3, 5, 20, 19 | ftc1cnnc 32654 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹)) |
150 | 119, 121,
136, 139, 11, 141 | dvmptntr 23540 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡))) |
151 | 22 | feqmptd 6159 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
152 | 149, 150,
151 | 3eqtr3d 2652 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
153 | 146, 137 | sylan2 490 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
154 | 116 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))) |
155 | 119, 121,
137, 139, 11, 141 | dvmptntr 23540 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))) |
156 | 154, 151,
155 | 3eqtr3rd 2653 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
157 | 144, 147,
148, 152, 153, 148, 156 | dvmptsub 23536 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)))) |
158 | 148 | subidd 10259 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0) |
159 | 158 | mpteq2dva 4672 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)) |
160 | 142, 157,
159 | 3eqtrd 2648 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)) |
161 | | fconstmpt 5085 |
. . . . . . . 8
⊢ ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0) |
162 | 160, 161 | syl6eqr 2662 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = ((𝐴(,)𝐵) × {0})) |
163 | 1, 3, 118, 162 | dveq0 23567 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) = ((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})) |
164 | 163 | fveq1d 6105 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵)) |
165 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵)) |
166 | | itgeq1 23345 |
. . . . . . . . 9
⊢ ((𝐴(,)𝑥) = (𝐴(,)𝐵) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
167 | 165, 166 | syl 17 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
168 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
169 | 167, 168 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
170 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) |
171 | | ovex 6577 |
. . . . . . 7
⊢
(∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) ∈ V |
172 | 169, 170,
171 | fvmpt 6191 |
. . . . . 6
⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
173 | 7, 172 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
174 | 164, 173 | eqtr3d 2646 |
. . . 4
⊢ (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
175 | | lbicc2 12159 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
176 | 2, 4, 5, 175 | syl3anc 1318 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
177 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴)) |
178 | | iooid 12074 |
. . . . . . . . . . 11
⊢ (𝐴(,)𝐴) = ∅ |
179 | 177, 178 | syl6eq 2660 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅) |
180 | | itgeq1 23345 |
. . . . . . . . . 10
⊢ ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡) |
181 | 179, 180 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡) |
182 | | itg0 23352 |
. . . . . . . . 9
⊢
∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0 |
183 | 181, 182 | syl6eq 2660 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0) |
184 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
185 | 183, 184 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (0 − (𝐹‘𝐴))) |
186 | | df-neg 10148 |
. . . . . . 7
⊢ -(𝐹‘𝐴) = (0 − (𝐹‘𝐴)) |
187 | 185, 186 | syl6eqr 2662 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = -(𝐹‘𝐴)) |
188 | | negex 10158 |
. . . . . 6
⊢ -(𝐹‘𝐴) ∈ V |
189 | 187, 170,
188 | fvmpt 6191 |
. . . . 5
⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴)) |
190 | 176, 189 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴)) |
191 | 10, 174, 190 | 3eqtr3d 2652 |
. . 3
⊢ (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) = -(𝐹‘𝐴)) |
192 | 191 | oveq2d 6565 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ((𝐹‘𝐵) + -(𝐹‘𝐴))) |
193 | 115, 7 | ffvelrnd 6268 |
. . 3
⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ) |
194 | 54 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
195 | 194, 57 | itgcl 23356 |
. . 3
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
196 | 193, 195 | pncan3d 10274 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
197 | 115, 176 | ffvelrnd 6268 |
. . 3
⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ) |
198 | 193, 197 | negsubd 10277 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + -(𝐹‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
199 | 192, 196,
198 | 3eqtr3d 2652 |
1
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |