Step | Hyp | Ref
| Expression |
1 | | ibladdnc.2 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) |
2 | | iblmbf 24512 |
. . . . . . . . 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 24381 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | | ibladdnc.4 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈
𝐿1) |
7 | | iblmbf 24512 |
. . . . . . . . 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 24381 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
11 | 5, 10 | readdd 14656 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 + 𝐶)) = ((ℜ‘𝐵) + (ℜ‘𝐶))) |
12 | 11 | itgeq2dv 24526 |
. . . . 5
⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥) |
13 | 5 | recld 14636 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ) |
14 | 5 | iblcn 24543 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1
∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1))) |
15 | 1, 14 | mpbid 235 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1)) |
16 | 15 | simpld 498 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈
𝐿1) |
17 | 10 | recld 14636 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℝ) |
18 | 10 | iblcn 24543 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿1
∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1))) |
19 | 6, 18 | mpbid 235 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1)) |
20 | 19 | simpld 498 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈
𝐿1) |
21 | 5, 10 | addcld 10731 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ) |
22 | | eqidd 2739 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) |
23 | | ref 14554 |
. . . . . . . . . . 11
⊢
ℜ:ℂ⟶ℝ |
24 | 23 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 →
ℜ:ℂ⟶ℝ) |
25 | 24 | feqmptd 6731 |
. . . . . . . . 9
⊢ (𝜑 → ℜ = (𝑦 ∈ ℂ ↦
(ℜ‘𝑦))) |
26 | | fveq2 6668 |
. . . . . . . . 9
⊢ (𝑦 = (𝐵 + 𝐶) → (ℜ‘𝑦) = (ℜ‘(𝐵 + 𝐶))) |
27 | 21, 22, 25, 26 | fmptco 6895 |
. . . . . . . 8
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 + 𝐶)))) |
28 | 11 | mpteq2dva 5122 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶)))) |
29 | 27, 28 | eqtrd 2773 |
. . . . . . 7
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶)))) |
30 | | ibladdnc.m |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) |
31 | 21 | fmpttd 6883 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ) |
32 | | ismbfcn 24374 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ↔ ((ℜ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn))) |
33 | 31, 32 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ↔ ((ℜ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn))) |
34 | 30, 33 | mpbid 235 |
. . . . . . . 8
⊢ (𝜑 → ((ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn)) |
35 | 34 | simpld 498 |
. . . . . . 7
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn) |
36 | 29, 35 | eqeltrrd 2834 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶))) ∈ MblFn) |
37 | 13, 16, 17, 20, 36, 13, 17 | itgaddnclem2 35448 |
. . . . 5
⊢ (𝜑 → ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥)) |
38 | 12, 37 | eqtrd 2773 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥)) |
39 | 5, 10 | imaddd 14657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐵 + 𝐶)) = ((ℑ‘𝐵) + (ℑ‘𝐶))) |
40 | 39 | itgeq2dv 24526 |
. . . . . . 7
⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥) |
41 | 5 | imcld 14637 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ) |
42 | 15 | simprd 499 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1) |
43 | 10 | imcld 14637 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℝ) |
44 | 19 | simprd 499 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1) |
45 | | imf 14555 |
. . . . . . . . . . . . 13
⊢
ℑ:ℂ⟶ℝ |
46 | 45 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 →
ℑ:ℂ⟶ℝ) |
47 | 46 | feqmptd 6731 |
. . . . . . . . . . 11
⊢ (𝜑 → ℑ = (𝑦 ∈ ℂ ↦
(ℑ‘𝑦))) |
48 | | fveq2 6668 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐵 + 𝐶) → (ℑ‘𝑦) = (ℑ‘(𝐵 + 𝐶))) |
49 | 21, 22, 47, 48 | fmptco 6895 |
. . . . . . . . . 10
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 + 𝐶)))) |
50 | 39 | mpteq2dva 5122 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶)))) |
51 | 49, 50 | eqtrd 2773 |
. . . . . . . . 9
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶)))) |
52 | 34 | simprd 499 |
. . . . . . . . 9
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn) |
53 | 51, 52 | eqeltrrd 2834 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶))) ∈ MblFn) |
54 | 41, 42, 43, 44, 53, 41, 43 | itgaddnclem2 35448 |
. . . . . . 7
⊢ (𝜑 → ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) |
55 | 40, 54 | eqtrd 2773 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) |
56 | 55 | oveq2d 7180 |
. . . . 5
⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥))) |
57 | | ax-icn 10667 |
. . . . . . 7
⊢ i ∈
ℂ |
58 | 57 | a1i 11 |
. . . . . 6
⊢ (𝜑 → i ∈
ℂ) |
59 | 41, 42 | itgcl 24528 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) |
60 | 43, 44 | itgcl 24528 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) |
61 | 58, 59, 60 | adddid 10736 |
. . . . 5
⊢ (𝜑 → (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
62 | 56, 61 | eqtrd 2773 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
63 | 38, 62 | oveq12d 7182 |
. . 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)))) |
64 | 13, 16 | itgcl 24528 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ) |
65 | 17, 20 | itgcl 24528 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘𝐶) d𝑥 ∈ ℂ) |
66 | | mulcl 10692 |
. . . . 5
⊢ ((i
∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i ·
∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ) |
67 | 57, 59, 66 | sylancr 590 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ) |
68 | | mulcl 10692 |
. . . . 5
⊢ ((i
∈ ℂ ∧ ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) → (i ·
∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ) |
69 | 57, 60, 68 | sylancr 590 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ) |
70 | 64, 65, 67, 69 | add4d 10939 |
. . 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)))) |
71 | 63, 70 | eqtrd 2773 |
. 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)))) |
72 | | ovexd 7199 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ V) |
73 | 4, 1, 9, 6, 30 | ibladdnc 35446 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈
𝐿1) |
74 | 72, 73 | itgcnval 24544 |
. 2
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥))) |
75 | 4, 1 | itgcnval 24544 |
. . 3
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥))) |
76 | 9, 6 | itgcnval 24544 |
. . 3
⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
77 | 75, 76 | oveq12d 7182 |
. 2
⊢ (𝜑 → (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))) |
78 | 71, 74, 77 | 3eqtr4d 2783 |
1
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) |