Step | Hyp | Ref
| Expression |
1 | | ftc2nc.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | 1 | rexrd 10281 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
3 | | ftc2nc.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
4 | 3 | rexrd 10281 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
5 | | ftc2nc.le |
. . . . . 6
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | | ubicc2 12482 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
7 | 2, 4, 5, 6 | syl3anc 1477 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
8 | | fvex 6362 |
. . . . . 6
⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) ∈ V |
9 | 8 | fvconst2 6633 |
. . . . 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 2760 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
12 | 11 | subcn 22870 |
. . . . . . . . 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 2760 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) |
15 | | ssid 3765 |
. . . . . . . . . 10
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵) |
16 | 15 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) |
17 | | ioossre 12428 |
. . . . . . . . . 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 22897 |
. . . . . . . . . 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 12464 |
. . . . . . . . . . . . 13
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
24 | | ffun 6209 |
. . . . . . . . . . . . 13
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → Fun (,)) |
25 | 23, 24 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
(,) |
26 | | fvelima 6410 |
. . . . . . . . . . . 12
⊢ ((Fun (,)
∧ 𝑠 ∈ ((,) “
((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠) |
27 | 25, 26 | mpan 708 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠) |
28 | | 1st2nd2 7372 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
29 | 28 | fveq2d 6356 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘〈(1st
‘𝑥), (2nd
‘𝑥)〉)) |
30 | | df-ov 6816 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘〈(1st
‘𝑥), (2nd
‘𝑥)〉) |
31 | 29, 30 | syl6eqr 2812 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥))) |
32 | 31 | eqeq1d 2762 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠)) |
33 | 32 | adantl 473 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠)) |
34 | 2, 4 | jca 555 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈
ℝ*)) |
35 | 34 | adantr 472 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈
ℝ*)) |
36 | | xp1st 7365 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st ‘𝑥) ∈ (𝐴[,]𝐵)) |
37 | | elicc1 12412 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((1st ‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵))) |
38 | 2, 4, 37 | syl2anc 696 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1st
‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵))) |
39 | 38 | biimpa 502 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (1st
‘𝑥) ∈ (𝐴[,]𝐵)) → ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵)) |
40 | 39 | simp2d 1138 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (1st
‘𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st ‘𝑥)) |
41 | 36, 40 | sylan2 492 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st ‘𝑥)) |
42 | | xp2nd 7366 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) |
43 | | iccleub 12422 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵) |
44 | 43 | 3expa 1112 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵) |
45 | 34, 42, 44 | syl2an 495 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd ‘𝑥) ≤ 𝐵) |
46 | | ioossioo 12458 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐴 ≤ (1st ‘𝑥) ∧ (2nd
‘𝑥) ≤ 𝐵)) → ((1st
‘𝑥)(,)(2nd
‘𝑥)) ⊆ (𝐴(,)𝐵)) |
47 | 35, 41, 45, 46 | syl12anc 1475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵)) |
48 | 47 | sselda 3744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 𝑡 ∈ (𝐴(,)𝐵)) |
49 | 22 | ffvelrnda 6522 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
50 | 49 | adantlr 753 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
51 | 48, 50 | syldan 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
52 | | ioombl 23533 |
. . . . . . . . . . . . . . . . . 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 | | fvexd 6364 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
55 | 22 | feqmptd 6411 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡))) |
56 | 55, 19 | eqeltrrd 2840 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
57 | 56 | adantr 472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
58 | 47, 53, 54, 57 | iblss 23770 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
59 | | ax-resscn 10185 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℝ
⊆ ℂ |
60 | | ssid 3765 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℂ
⊆ ℂ |
61 | | cncfss 22903 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)) |
62 | 59, 60, 61 | mp2an 710 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(ℂ–cn→ℝ)
⊆ (ℂ–cn→ℂ) |
63 | | abscncf 22905 |
. . . . . . . . . . . . . . . . . . . 20
⊢ abs
∈ (ℂ–cn→ℝ) |
64 | 62, 63 | sselii 3741 |
. . . . . . . . . . . . . . . . . . 19
⊢ abs
∈ (ℂ–cn→ℂ) |
65 | 64 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ)) |
66 | 55 | reseq1d 5550 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((1st
‘𝑥)(,)(2nd
‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥)))) |
67 | 66 | adantr 472 |
. . . . . . . . . . . . . . . . . . . 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)))) |
68 | 47 | resmptd 5610 |
. . . . . . . . . . . . . . . . . . . 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))) |
69 | 67, 68 | eqtrd 2794 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡))) |
70 | 20 | adantr 472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
71 | | rescncf 22901 |
. . . . . . . . . . . . . . . . . . . 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))) |
72 | 47, 70, 71 | sylc 65 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) |
73 | 69, 72 | eqeltrrd 2840 |
. . . . . . . . . . . . . . . . . 18
⊢ ((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 22917 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) |
75 | | cnmbf 23625 |
. . . . . . . . . . . . . . . . 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) |
76 | 52, 74, 75 | sylancr 698 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
MblFn) |
77 | 51, 58 | itgcl 23749 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
78 | 77 | cjcld 14135 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) →
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ) |
79 | | ioossre 12428 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℝ |
80 | 79, 59 | sstri 3753 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℂ |
81 | | cncfmptc 22915 |
. . . . . . . . . . . . . . . . . . . 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)) |
82 | 80, 60, 81 | mp3an23 1565 |
. . . . . . . . . . . . . . . . . . 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)) |
83 | 78, 82 | 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)) |
84 | | nfcv 2902 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠((ℝ D 𝐹)‘𝑡) |
85 | | nfcsb1v 3690 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡) |
86 | | csbeq1a 3683 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) |
87 | 84, 85, 86 | cbvmpt 4901 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)) ↦
((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) |
88 | 87, 73 | syl5eqelr 2844 |
. . . . . . . . . . . . . . . . . 18
⊢ ((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 23415 |
. . . . . . . . . . . . . . . . 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)) |
90 | | cnmbf 23625 |
. . . . . . . . . . . . . . . . 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) |
91 | 52, 89, 90 | sylancr 698 |
. . . . . . . . . . . . . . . 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) |
92 | 51, 58, 76, 91 | itgabsnc 33792 |
. . . . . . . . . . . . . . 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) |
93 | 51 | abscld 14374 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → (abs‘((ℝ
D 𝐹)‘𝑡)) ∈
ℝ) |
94 | | fvexd 6364 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
95 | 94, 58, 76 | iblabsnc 33787 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
𝐿1) |
96 | 51 | absge0d 14382 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 0 ≤
(abs‘((ℝ D 𝐹)‘𝑡))) |
97 | 93, 95, 96 | itgposval 23761 |
. . . . . . . . . . . . . . 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)))) |
98 | 92, 97 | breqtrd 4830 |
. . . . . . . . . . . . . 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)))) |
99 | | itgeq1 23738 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) |
100 | 99 | fveq2d 6356 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)) |
101 | | eleq2 2828 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↔ 𝑡 ∈ 𝑠)) |
102 | 101 | ifbid 4252 |
. . . . . . . . . . . . . . . . 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)) |
103 | 102 | mpteq2dv 4897 |
. . . . . . . . . . . . . . . 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))) |
104 | 103 | fveq2d 6356 |
. . . . . . . . . . . . . . 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)))) |
105 | 100, 104 | breq12d 4817 |
. . . . . . . . . . . . . 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))))) |
106 | 98, 105 | syl5ibcom 235 |
. . . . . . . . . . . . 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))))) |
107 | 33, 106 | sylbid 230 |
. . . . . . . . . . . 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))))) |
108 | 107 | rexlimdva 3169 |
. . . . . . . . . . 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))))) |
109 | 27, 108 | syl5 34 |
. . . . . . . . . 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))))) |
110 | 109 | ralrimiv 3103 |
. . . . . . . . 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)))) |
111 | 14, 1, 3, 5, 16, 18, 19, 22, 110 | ftc1anc 33806 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
112 | | ftc2nc.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
113 | | cncff 22897 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
114 | 112, 113 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
115 | 114 | feqmptd 6411 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) |
116 | 115, 112 | eqeltrrd 2840 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
117 | 11, 13, 111, 116 | cncfmpt2f 22918 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
118 | 59 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
119 | | iccssre 12448 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
120 | 1, 3, 119 | syl2anc 696 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
121 | | fvexd 6364 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
122 | 3 | adantr 472 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
123 | 122 | rexrd 10281 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈
ℝ*) |
124 | | elicc2 12431 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
125 | 1, 3, 124 | syl2anc 696 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
126 | 125 | biimpa 502 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
127 | 126 | simp3d 1139 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) |
128 | | iooss2 12404 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℝ*
∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
129 | 123, 127,
128 | syl2anc 696 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
130 | | ioombl 23533 |
. . . . . . . . . . . . . 14
⊢ (𝐴(,)𝑥) ∈ dom vol |
131 | 130 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol) |
132 | | fvexd 6364 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
133 | 56 | adantr 472 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
134 | 129, 131,
132, 133 | iblss 23770 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
135 | 121, 134 | itgcl 23749 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
136 | 114 | ffvelrnda 6522 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
137 | 135, 136 | subcld 10584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) ∈ ℂ) |
138 | 11 | tgioo2 22807 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
139 | | iccntr 22825 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
140 | 1, 3, 139 | syl2anc 696 |
. . . . . . . . . 10
⊢ (𝜑 →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
141 | 118, 120,
137, 138, 11, 140 | dvmptntr 23933 |
. . . . . . . . 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))))) |
142 | | reelprrecn 10220 |
. . . . . . . . . . 11
⊢ ℝ
∈ {ℝ, ℂ} |
143 | 142 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
144 | | ioossicc 12452 |
. . . . . . . . . . . 12
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
145 | 144 | sseli 3740 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
146 | 145, 135 | sylan2 492 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
147 | 22 | ffvelrnda 6522 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ) |
148 | 14, 1, 3, 5, 20, 19 | ftc1cnnc 33797 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹)) |
149 | 118, 120,
135, 138, 11, 140 | dvmptntr 23933 |
. . . . . . . . . . 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))) |
150 | 22 | feqmptd 6411 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
151 | 148, 149,
150 | 3eqtr3d 2802 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
152 | 145, 136 | sylan2 492 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
153 | 115 | oveq2d 6829 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))) |
154 | 118, 120,
136, 138, 11, 140 | dvmptntr 23933 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))) |
155 | 153, 150,
154 | 3eqtr3rd 2803 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
156 | 143, 146,
147, 151, 152, 147, 155 | dvmptsub 23929 |
. . . . . . . . 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)))) |
157 | 147 | subidd 10572 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0) |
158 | 157 | mpteq2dva 4896 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)) |
159 | 141, 156,
158 | 3eqtrd 2798 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)) |
160 | | fconstmpt 5320 |
. . . . . . . 8
⊢ ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0) |
161 | 159, 160 | syl6eqr 2812 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = ((𝐴(,)𝐵) × {0})) |
162 | 1, 3, 117, 161 | dveq0 23962 |
. . . . . 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)})) |
163 | 162 | fveq1d 6354 |
. . . . 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)) |
164 | | oveq2 6821 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵)) |
165 | | itgeq1 23738 |
. . . . . . . . 9
⊢ ((𝐴(,)𝑥) = (𝐴(,)𝐵) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
166 | 164, 165 | syl 17 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
167 | | fveq2 6352 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
168 | 166, 167 | oveq12d 6831 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
169 | | eqid 2760 |
. . . . . . 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))) |
170 | | ovex 6841 |
. . . . . . 7
⊢
(∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) ∈ V |
171 | 168, 169,
170 | fvmpt 6444 |
. . . . . 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))) |
172 | 7, 171 | 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))) |
173 | 163, 172 | eqtr3d 2796 |
. . . 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))) |
174 | | lbicc2 12481 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
175 | 2, 4, 5, 174 | syl3anc 1477 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
176 | | oveq2 6821 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴)) |
177 | | iooid 12396 |
. . . . . . . . . . 11
⊢ (𝐴(,)𝐴) = ∅ |
178 | 176, 177 | syl6eq 2810 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅) |
179 | | itgeq1 23738 |
. . . . . . . . . 10
⊢ ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡) |
180 | 178, 179 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡) |
181 | | itg0 23745 |
. . . . . . . . 9
⊢
∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0 |
182 | 180, 181 | syl6eq 2810 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0) |
183 | | fveq2 6352 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
184 | 182, 183 | oveq12d 6831 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (0 − (𝐹‘𝐴))) |
185 | | df-neg 10461 |
. . . . . . 7
⊢ -(𝐹‘𝐴) = (0 − (𝐹‘𝐴)) |
186 | 184, 185 | syl6eqr 2812 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = -(𝐹‘𝐴)) |
187 | | negex 10471 |
. . . . . 6
⊢ -(𝐹‘𝐴) ∈ V |
188 | 186, 169,
187 | fvmpt 6444 |
. . . . 5
⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴)) |
189 | 175, 188 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴)) |
190 | 10, 173, 189 | 3eqtr3d 2802 |
. . 3
⊢ (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) = -(𝐹‘𝐴)) |
191 | 190 | oveq2d 6829 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ((𝐹‘𝐵) + -(𝐹‘𝐴))) |
192 | 114, 7 | ffvelrnd 6523 |
. . 3
⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ) |
193 | | fvexd 6364 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
194 | 193, 56 | itgcl 23749 |
. . 3
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
195 | 192, 194 | pncan3d 10587 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
196 | 114, 175 | ffvelrnd 6523 |
. . 3
⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ) |
197 | 192, 196 | negsubd 10590 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + -(𝐹‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
198 | 191, 195,
197 | 3eqtr3d 2802 |
1
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |