Step | Hyp | Ref
| Expression |
1 | | ibladdnc.2 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) |
2 | | iblmbf 23534 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
3 | 1, 2 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
4 | | ibladdnc.1 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
5 | 3, 4 | mbfmptcl 23404 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | | ibladdnc.4 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈
𝐿1) |
7 | | iblmbf 23534 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) |
8 | 6, 7 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) |
9 | | ibladdnc.3 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
10 | 8, 9 | mbfmptcl 23404 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
11 | 5, 10 | readdd 13954 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 + 𝐶)) = ((ℜ‘𝐵) + (ℜ‘𝐶))) |
12 | 11 | itgeq2dv 23548 |
. . . . 5
⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥) |
13 | 5 | recld 13934 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ) |
14 | 5 | iblcn 23565 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1
∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1))) |
15 | 1, 14 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1)) |
16 | 15 | simpld 475 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈
𝐿1) |
17 | 10 | recld 13934 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℝ) |
18 | 10 | iblcn 23565 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿1
∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1))) |
19 | 6, 18 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1)) |
20 | 19 | simpld 475 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈
𝐿1) |
21 | 5, 10 | addcld 10059 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ) |
22 | | eqidd 2623 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) |
23 | | ref 13852 |
. . . . . . . . . . 11
⊢
ℜ:ℂ⟶ℝ |
24 | 23 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 →
ℜ:ℂ⟶ℝ) |
25 | 24 | feqmptd 6249 |
. . . . . . . . 9
⊢ (𝜑 → ℜ = (𝑦 ∈ ℂ ↦
(ℜ‘𝑦))) |
26 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑦 = (𝐵 + 𝐶) → (ℜ‘𝑦) = (ℜ‘(𝐵 + 𝐶))) |
27 | 21, 22, 25, 26 | fmptco 6396 |
. . . . . . . 8
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 + 𝐶)))) |
28 | 11 | mpteq2dva 4744 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶)))) |
29 | 27, 28 | eqtrd 2656 |
. . . . . . 7
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶)))) |
30 | | ibladdnc.m |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) |
31 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) |
32 | 21, 31 | fmptd 6385 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ) |
33 | | ismbfcn 23398 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ↔ ((ℜ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn))) |
34 | 32, 33 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ↔ ((ℜ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn))) |
35 | 30, 34 | mpbid 222 |
. . . . . . . 8
⊢ (𝜑 → ((ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn)) |
36 | 35 | simpld 475 |
. . . . . . 7
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn) |
37 | 29, 36 | eqeltrrd 2702 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶))) ∈ MblFn) |
38 | 13, 16, 17, 20, 37, 13, 17 | itgaddnclem2 33469 |
. . . . 5
⊢ (𝜑 → ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥)) |
39 | 12, 38 | eqtrd 2656 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥)) |
40 | 5, 10 | imaddd 13955 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐵 + 𝐶)) = ((ℑ‘𝐵) + (ℑ‘𝐶))) |
41 | 40 | itgeq2dv 23548 |
. . . . . . 7
⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥) |
42 | 5 | imcld 13935 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ) |
43 | 15 | simprd 479 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1) |
44 | 10 | imcld 13935 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℝ) |
45 | 19 | simprd 479 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1) |
46 | | imf 13853 |
. . . . . . . . . . . . 13
⊢
ℑ:ℂ⟶ℝ |
47 | 46 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 →
ℑ:ℂ⟶ℝ) |
48 | 47 | feqmptd 6249 |
. . . . . . . . . . 11
⊢ (𝜑 → ℑ = (𝑦 ∈ ℂ ↦
(ℑ‘𝑦))) |
49 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐵 + 𝐶) → (ℑ‘𝑦) = (ℑ‘(𝐵 + 𝐶))) |
50 | 21, 22, 48, 49 | fmptco 6396 |
. . . . . . . . . 10
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 + 𝐶)))) |
51 | 40 | mpteq2dva 4744 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶)))) |
52 | 50, 51 | eqtrd 2656 |
. . . . . . . . 9
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶)))) |
53 | 35 | simprd 479 |
. . . . . . . . 9
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn) |
54 | 52, 53 | eqeltrrd 2702 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶))) ∈ MblFn) |
55 | 42, 43, 44, 45, 54, 42, 44 | itgaddnclem2 33469 |
. . . . . . 7
⊢ (𝜑 → ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) |
56 | 41, 55 | eqtrd 2656 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) |
57 | 56 | oveq2d 6666 |
. . . . 5
⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥))) |
58 | | ax-icn 9995 |
. . . . . . 7
⊢ i ∈
ℂ |
59 | 58 | a1i 11 |
. . . . . 6
⊢ (𝜑 → i ∈
ℂ) |
60 | 42, 43 | itgcl 23550 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) |
61 | 44, 45 | itgcl 23550 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) |
62 | 59, 60, 61 | adddid 10064 |
. . . . 5
⊢ (𝜑 → (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
63 | 57, 62 | eqtrd 2656 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
64 | 39, 63 | oveq12d 6668 |
. . 3
⊢ (𝜑 → (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥)) = ((∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥) + ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))) |
65 | 13, 16 | itgcl 23550 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ) |
66 | 17, 20 | itgcl 23550 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘𝐶) d𝑥 ∈ ℂ) |
67 | | mulcl 10020 |
. . . . 5
⊢ ((i
∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i ·
∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ) |
68 | 58, 60, 67 | sylancr 695 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ) |
69 | | mulcl 10020 |
. . . . 5
⊢ ((i
∈ ℂ ∧ ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) → (i ·
∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ) |
70 | 58, 61, 69 | sylancr 695 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ) |
71 | 65, 66, 68, 70 | add4d 10264 |
. . 3
⊢ (𝜑 → ((∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥) + ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))) |
72 | 64, 71 | eqtrd 2656 |
. 2
⊢ (𝜑 → (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥)) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))) |
73 | | ovexd 6680 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ V) |
74 | 4, 1, 9, 6, 30 | ibladdnc 33467 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈
𝐿1) |
75 | 73, 74 | itgcnval 23566 |
. 2
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥))) |
76 | 4, 1 | itgcnval 23566 |
. . 3
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥))) |
77 | 9, 6 | itgcnval 23566 |
. . 3
⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
78 | 76, 77 | oveq12d 6668 |
. 2
⊢ (𝜑 → (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))) |
79 | 72, 75, 78 | 3eqtr4d 2666 |
1
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) |