Step | Hyp | Ref
| Expression |
1 | | itgpowd.4 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
2 | | nn0p1nn 11332 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → (𝑁 + 1) ∈ ℕ) |
4 | 3 | nncnd 11036 |
. 2
⊢ (𝜑 → (𝑁 + 1) ∈ ℂ) |
5 | | itgpowd.1 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
6 | | itgpowd.2 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
7 | | iccssre 12255 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
8 | 5, 6, 7 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
9 | | ax-resscn 9993 |
. . . . . 6
⊢ ℝ
⊆ ℂ |
10 | 8, 9 | syl6ss 3615 |
. . . . 5
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ) |
11 | 10 | sselda 3603 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ) |
12 | 1 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑁 ∈
ℕ0) |
13 | 11, 12 | expcld 13008 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥↑𝑁) ∈ ℂ) |
14 | 10 | resmptd 5452 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝑥↑𝑁))) |
15 | | expcncf 22725 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ (𝑥 ∈ ℂ
↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) |
16 | 1, 15 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) |
17 | | rescncf 22700 |
. . . . . 6
⊢ ((𝐴[,]𝐵) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))) |
18 | 10, 16, 17 | sylc 65 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
19 | 14, 18 | eqeltrrd 2702 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝑥↑𝑁)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
20 | | cniccibl 23607 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝑥↑𝑁)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝑥↑𝑁)) ∈
𝐿1) |
21 | 5, 6, 19, 20 | syl3anc 1326 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝑥↑𝑁)) ∈
𝐿1) |
22 | 13, 21 | itgcl 23550 |
. 2
⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥 ∈ ℂ) |
23 | 3 | nnne0d 11065 |
. 2
⊢ (𝜑 → (𝑁 + 1) ≠ 0) |
24 | 4, 13, 21 | itgmulc2 23600 |
. . 3
⊢ (𝜑 → ((𝑁 + 1) · ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥) = ∫(𝐴[,]𝐵)((𝑁 + 1) · (𝑥↑𝑁)) d𝑥) |
25 | | eqidd 2623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
26 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑡 = 𝑥 → (𝑡↑𝑁) = (𝑥↑𝑁)) |
27 | 26 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑡 = 𝑥 → ((𝑁 + 1) · (𝑡↑𝑁)) = ((𝑁 + 1) · (𝑥↑𝑁))) |
28 | 27 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑡 = 𝑥) → ((𝑁 + 1) · (𝑡↑𝑁)) = ((𝑁 + 1) · (𝑥↑𝑁))) |
29 | | simpr 477 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵)) |
30 | 4 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 + 1) ∈ ℂ) |
31 | | ioossicc 12259 |
. . . . . . . . . 10
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
32 | 31 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
33 | 32 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
34 | 33, 13 | syldan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥↑𝑁) ∈ ℂ) |
35 | 30, 34 | mulcld 10060 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 + 1) · (𝑥↑𝑁)) ∈ ℂ) |
36 | 25, 28, 29, 35 | fvmptd 6288 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥) = ((𝑁 + 1) · (𝑥↑𝑁))) |
37 | 36 | itgeq2dv 23548 |
. . . 4
⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥) d𝑥 = ∫(𝐴(,)𝐵)((𝑁 + 1) · (𝑥↑𝑁)) d𝑥) |
38 | | itgpowd.3 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
39 | | reelprrecn 10028 |
. . . . . . . . 9
⊢ ℝ
∈ {ℝ, ℂ} |
40 | 39 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
41 | 9 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
42 | 41 | sselda 3603 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ) |
43 | | 1nn0 11308 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ0 |
44 | 43 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℕ0) |
45 | 1, 44 | nn0addcld 11355 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 + 1) ∈
ℕ0) |
46 | 45 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑁 + 1) ∈
ℕ0) |
47 | 42, 46 | expcld 13008 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑡↑(𝑁 + 1)) ∈ ℂ) |
48 | 1 | nn0cnd 11353 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℂ) |
49 | 48 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑁 ∈ ℂ) |
50 | | 1cnd 10056 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 1 ∈
ℂ) |
51 | 49, 50 | addcld 10059 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑁 + 1) ∈ ℂ) |
52 | 1 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑁 ∈
ℕ0) |
53 | 42, 52 | expcld 13008 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (𝑡↑𝑁) ∈ ℂ) |
54 | 51, 53 | mulcld 10060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ((𝑁 + 1) · (𝑡↑𝑁)) ∈ ℂ) |
55 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → 𝑡 ∈ ℂ) |
56 | 45 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑁 + 1) ∈
ℕ0) |
57 | 55, 56 | expcld 13008 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑡↑(𝑁 + 1)) ∈ ℂ) |
58 | | eqid 2622 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) = (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) |
59 | 57, 58 | fmptd 6385 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 +
1))):ℂ⟶ℂ) |
60 | | ssid 3624 |
. . . . . . . . . . . 12
⊢ ℂ
⊆ ℂ |
61 | 60 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → ℂ ⊆
ℂ) |
62 | 4 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑁 + 1) ∈ ℂ) |
63 | 1 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → 𝑁 ∈
ℕ0) |
64 | 55, 63 | expcld 13008 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → (𝑡↑𝑁) ∈ ℂ) |
65 | 62, 64 | mulcld 10060 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑡 ∈ ℂ) → ((𝑁 + 1) · (𝑡↑𝑁)) ∈ ℂ) |
66 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁))) = (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁))) |
67 | 65, 66 | fmptd 6385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁))):ℂ⟶ℂ) |
68 | | dvexp 23716 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 + 1) ∈ ℕ →
(ℂ D (𝑡 ∈
ℂ ↦ (𝑡↑(𝑁 + 1)))) = (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑((𝑁 + 1) − 1))))) |
69 | 3, 68 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1)))) = (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑((𝑁 + 1) − 1))))) |
70 | | 1cnd 10056 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 1 ∈
ℂ) |
71 | 48, 70 | pncand 10393 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
72 | 71 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑡↑((𝑁 + 1) − 1)) = (𝑡↑𝑁)) |
73 | 72 | oveq2d 6666 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑁 + 1) · (𝑡↑((𝑁 + 1) − 1))) = ((𝑁 + 1) · (𝑡↑𝑁))) |
74 | 73 | mpteq2dv 4745 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑((𝑁 + 1) − 1)))) = (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
75 | 69, 74 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1)))) = (𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
76 | 75 | feq1d 6030 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1)))):ℂ⟶ℂ ↔
(𝑡 ∈ ℂ ↦
((𝑁 + 1) · (𝑡↑𝑁))):ℂ⟶ℂ)) |
77 | 67, 76 | mpbird 247 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 +
1)))):ℂ⟶ℂ) |
78 | | fdm 6051 |
. . . . . . . . . . . . 13
⊢ ((ℂ
D (𝑡 ∈ ℂ ↦
(𝑡↑(𝑁 + 1)))):ℂ⟶ℂ → dom
(ℂ D (𝑡 ∈
ℂ ↦ (𝑡↑(𝑁 + 1)))) = ℂ) |
79 | 77, 78 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom (ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1)))) = ℂ) |
80 | 9, 79 | syl5sseqr 3654 |
. . . . . . . . . . 11
⊢ (𝜑 → ℝ ⊆ dom
(ℂ D (𝑡 ∈
ℂ ↦ (𝑡↑(𝑁 + 1))))) |
81 | | dvres3 23677 |
. . . . . . . . . . 11
⊢
(((ℝ ∈ {ℝ, ℂ} ∧ (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))):ℂ⟶ℂ) ∧
(ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1)))))) → (ℝ D ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ ℝ)) = ((ℂ D
(𝑡 ∈ ℂ ↦
(𝑡↑(𝑁 + 1)))) ↾ ℝ)) |
82 | 40, 59, 61, 80, 81 | syl22anc 1327 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ ℝ)) = ((ℂ D
(𝑡 ∈ ℂ ↦
(𝑡↑(𝑁 + 1)))) ↾ ℝ)) |
83 | 75 | reseq1d 5395 |
. . . . . . . . . 10
⊢ (𝜑 → ((ℂ D (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1)))) ↾ ℝ) = ((𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ↾ ℝ)) |
84 | 82, 83 | eqtrd 2656 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ ℝ)) = ((𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ↾ ℝ)) |
85 | | resmpt 5449 |
. . . . . . . . . . 11
⊢ (ℝ
⊆ ℂ → ((𝑡
∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ ℝ) = (𝑡 ∈ ℝ ↦ (𝑡↑(𝑁 + 1)))) |
86 | 9, 85 | mp1i 13 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ ℝ) = (𝑡 ∈ ℝ ↦ (𝑡↑(𝑁 + 1)))) |
87 | 86 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ ℝ)) = (ℝ D
(𝑡 ∈ ℝ ↦
(𝑡↑(𝑁 + 1))))) |
88 | | resmpt 5449 |
. . . . . . . . . 10
⊢ (ℝ
⊆ ℂ → ((𝑡
∈ ℂ ↦ ((𝑁
+ 1) · (𝑡↑𝑁))) ↾ ℝ) = (𝑡 ∈ ℝ ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
89 | 9, 88 | mp1i 13 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ↾ ℝ) = (𝑡 ∈ ℝ ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
90 | 84, 87, 89 | 3eqtr3d 2664 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ (𝑡↑(𝑁 + 1)))) = (𝑡 ∈ ℝ ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
91 | | eqid 2622 |
. . . . . . . . 9
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
92 | 91 | tgioo2 22606 |
. . . . . . . 8
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
93 | | iccntr 22624 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
94 | 5, 6, 93 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
95 | 40, 47, 54, 90, 8, 92, 91, 94 | dvmptres2 23725 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))) |
96 | | ioossre 12235 |
. . . . . . . . . . 11
⊢ (𝐴(,)𝐵) ⊆ ℝ |
97 | 96, 9 | sstri 3612 |
. . . . . . . . . 10
⊢ (𝐴(,)𝐵) ⊆ ℂ |
98 | 97 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
99 | | cncfmptc 22714 |
. . . . . . . . 9
⊢ (((𝑁 + 1) ∈ ℂ ∧
(𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆
ℂ) → (𝑡 ∈
(𝐴(,)𝐵) ↦ (𝑁 + 1)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
100 | 4, 98, 61, 99 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑁 + 1)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
101 | | resmpt 5449 |
. . . . . . . . . 10
⊢ ((𝐴(,)𝐵) ⊆ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡↑𝑁))) |
102 | 97, 101 | mp1i 13 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡↑𝑁))) |
103 | | expcncf 22725 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ (𝑡 ∈ ℂ
↦ (𝑡↑𝑁)) ∈ (ℂ–cn→ℂ)) |
104 | 1, 103 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ∈ (ℂ–cn→ℂ)) |
105 | | rescncf 22700 |
. . . . . . . . . 10
⊢ ((𝐴(,)𝐵) ⊆ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ∈ (ℂ–cn→ℂ) → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))) |
106 | 98, 104, 105 | sylc 65 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
107 | 102, 106 | eqeltrrd 2702 |
. . . . . . . 8
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡↑𝑁)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
108 | 100, 107 | mulcncf 23215 |
. . . . . . 7
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
109 | 95, 108 | eqeltrd 2701 |
. . . . . 6
⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
110 | | ioombl 23333 |
. . . . . . . . 9
⊢ (𝐴(,)𝐵) ∈ dom vol |
111 | 110 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol) |
112 | 48 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑁 ∈ ℂ) |
113 | | 1cnd 10056 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 1 ∈ ℂ) |
114 | 112, 113 | addcld 10059 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑁 + 1) ∈ ℂ) |
115 | 10 | sselda 3603 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℂ) |
116 | 1 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑁 ∈
ℕ0) |
117 | 115, 116 | expcld 13008 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡↑𝑁) ∈ ℂ) |
118 | 114, 117 | mulcld 10060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → ((𝑁 + 1) · (𝑡↑𝑁)) ∈ ℂ) |
119 | | cncfmptc 22714 |
. . . . . . . . . . 11
⊢ (((𝑁 + 1) ∈ ℂ ∧
(𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆
ℂ) → (𝑡 ∈
(𝐴[,]𝐵) ↦ (𝑁 + 1)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
120 | 4, 10, 61, 119 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑁 + 1)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
121 | 10 | resmptd 5452 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑𝑁))) |
122 | | rescncf 22700 |
. . . . . . . . . . . 12
⊢ ((𝐴[,]𝐵) ⊆ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ∈ (ℂ–cn→ℂ) → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))) |
123 | 10, 104, 122 | sylc 65 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑𝑁)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
124 | 121, 123 | eqeltrrd 2702 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑𝑁)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
125 | 120, 124 | mulcncf 23215 |
. . . . . . . . 9
⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
126 | | cniccibl 23607 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑡 ∈ (𝐴[,]𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑡 ∈ (𝐴[,]𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ∈
𝐿1) |
127 | 5, 6, 125, 126 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ∈
𝐿1) |
128 | 32, 111, 118, 127 | iblss 23571 |
. . . . . . 7
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁))) ∈
𝐿1) |
129 | 95, 128 | eqeltrd 2701 |
. . . . . 6
⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))) ∈
𝐿1) |
130 | 10 | resmptd 5452 |
. . . . . . 7
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))) |
131 | | expcncf 22725 |
. . . . . . . . 9
⊢ ((𝑁 + 1) ∈ ℕ0
→ (𝑡 ∈ ℂ
↦ (𝑡↑(𝑁 + 1))) ∈
(ℂ–cn→ℂ)) |
132 | 45, 131 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ∈ (ℂ–cn→ℂ)) |
133 | | rescncf 22700 |
. . . . . . . 8
⊢ ((𝐴[,]𝐵) ⊆ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ∈ (ℂ–cn→ℂ) → ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))) |
134 | 10, 132, 133 | sylc 65 |
. . . . . . 7
⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡↑(𝑁 + 1))) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
135 | 130, 134 | eqeltrrd 2702 |
. . . . . 6
⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
136 | 5, 6, 38, 109, 129, 135 | ftc2 23807 |
. . . . 5
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))))‘𝑥) d𝑥 = (((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))‘𝐵) − ((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))‘𝐴))) |
137 | 95 | fveq1d 6193 |
. . . . . . 7
⊢ (𝜑 → ((ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))))‘𝑥) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥)) |
138 | 137 | ralrimivw 2967 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)((ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))))‘𝑥) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥)) |
139 | | itgeq2 23544 |
. . . . . 6
⊢
(∀𝑥 ∈
(𝐴(,)𝐵)((ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))))‘𝑥) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥) → ∫(𝐴(,)𝐵)((ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))))‘𝑥) d𝑥 = ∫(𝐴(,)𝐵)((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥) d𝑥) |
140 | 138, 139 | syl 17 |
. . . . 5
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))))‘𝑥) d𝑥 = ∫(𝐴(,)𝐵)((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥) d𝑥) |
141 | | eqidd 2623 |
. . . . . . 7
⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1))) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))) |
142 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 = 𝐵) → 𝑡 = 𝐵) |
143 | 142 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 = 𝐵) → (𝑡↑(𝑁 + 1)) = (𝐵↑(𝑁 + 1))) |
144 | 5 | rexrd 10089 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
145 | 6 | rexrd 10089 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
146 | | ubicc2 12289 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
147 | 144, 145,
38, 146 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
148 | 6 | recnd 10068 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℂ) |
149 | 148, 45 | expcld 13008 |
. . . . . . 7
⊢ (𝜑 → (𝐵↑(𝑁 + 1)) ∈ ℂ) |
150 | 141, 143,
147, 149 | fvmptd 6288 |
. . . . . 6
⊢ (𝜑 → ((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))‘𝐵) = (𝐵↑(𝑁 + 1))) |
151 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 = 𝐴) → 𝑡 = 𝐴) |
152 | 151 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 = 𝐴) → (𝑡↑(𝑁 + 1)) = (𝐴↑(𝑁 + 1))) |
153 | | lbicc2 12288 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
154 | 144, 145,
38, 153 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
155 | 5 | recnd 10068 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℂ) |
156 | 155, 45 | expcld 13008 |
. . . . . . 7
⊢ (𝜑 → (𝐴↑(𝑁 + 1)) ∈ ℂ) |
157 | 141, 152,
154, 156 | fvmptd 6288 |
. . . . . 6
⊢ (𝜑 → ((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))‘𝐴) = (𝐴↑(𝑁 + 1))) |
158 | 150, 157 | oveq12d 6668 |
. . . . 5
⊢ (𝜑 → (((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))‘𝐵) − ((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡↑(𝑁 + 1)))‘𝐴)) = ((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1)))) |
159 | 136, 140,
158 | 3eqtr3d 2664 |
. . . 4
⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝑡 ∈ (𝐴(,)𝐵) ↦ ((𝑁 + 1) · (𝑡↑𝑁)))‘𝑥) d𝑥 = ((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1)))) |
160 | 4 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑁 + 1) ∈ ℂ) |
161 | 160, 13 | mulcld 10060 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝑁 + 1) · (𝑥↑𝑁)) ∈ ℂ) |
162 | 5, 6, 161 | itgioo 23582 |
. . . 4
⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝑁 + 1) · (𝑥↑𝑁)) d𝑥 = ∫(𝐴[,]𝐵)((𝑁 + 1) · (𝑥↑𝑁)) d𝑥) |
163 | 37, 159, 162 | 3eqtr3rd 2665 |
. . 3
⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝑁 + 1) · (𝑥↑𝑁)) d𝑥 = ((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1)))) |
164 | 24, 163 | eqtrd 2656 |
. 2
⊢ (𝜑 → ((𝑁 + 1) · ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥) = ((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1)))) |
165 | 4, 22, 23, 164 | mvllmuld 10857 |
1
⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1))) |