| Step | Hyp | Ref
| Expression |
| 1 | | ibladdnc.2 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) |
| 2 | | iblmbf 25802 |
. . . . . . . . 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 25671 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
| 6 | | ibladdnc.4 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈
𝐿1) |
| 7 | | iblmbf 25802 |
. . . . . . . . 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 25671 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 11 | 5, 10 | readdd 15253 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 + 𝐶)) = ((ℜ‘𝐵) + (ℜ‘𝐶))) |
| 12 | 11 | itgeq2dv 25817 |
. . . . 5
⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥) |
| 13 | 5 | recld 15233 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ) |
| 14 | 5 | iblcn 25834 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1
∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1))) |
| 15 | 1, 14 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1)) |
| 16 | 15 | simpld 494 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈
𝐿1) |
| 17 | 10 | recld 15233 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℝ) |
| 18 | 10 | iblcn 25834 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿1
∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1))) |
| 19 | 6, 18 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1)) |
| 20 | 19 | simpld 494 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈
𝐿1) |
| 21 | 5, 10 | addcld 11280 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ) |
| 22 | | eqidd 2738 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) |
| 23 | | ref 15151 |
. . . . . . . . . . 11
⊢
ℜ:ℂ⟶ℝ |
| 24 | 23 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 →
ℜ:ℂ⟶ℝ) |
| 25 | 24 | feqmptd 6977 |
. . . . . . . . 9
⊢ (𝜑 → ℜ = (𝑦 ∈ ℂ ↦
(ℜ‘𝑦))) |
| 26 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑦 = (𝐵 + 𝐶) → (ℜ‘𝑦) = (ℜ‘(𝐵 + 𝐶))) |
| 27 | 21, 22, 25, 26 | fmptco 7149 |
. . . . . . . 8
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 + 𝐶)))) |
| 28 | 11 | mpteq2dva 5242 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶)))) |
| 29 | 27, 28 | eqtrd 2777 |
. . . . . . 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 7135 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ) |
| 32 | | ismbfcn 25664 |
. . . . . . . . . 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 232 |
. . . . . . . 8
⊢ (𝜑 → ((ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn ∧ (ℑ ∘
(𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn)) |
| 35 | 34 | simpld 494 |
. . . . . . 7
⊢ (𝜑 → (ℜ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn) |
| 36 | 29, 35 | eqeltrrd 2842 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐵) + (ℜ‘𝐶))) ∈ MblFn) |
| 37 | 13, 16, 17, 20, 36, 13, 17 | itgaddnclem2 37686 |
. . . . 5
⊢ (𝜑 → ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥)) |
| 38 | 12, 37 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥)) |
| 39 | 5, 10 | imaddd 15254 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐵 + 𝐶)) = ((ℑ‘𝐵) + (ℑ‘𝐶))) |
| 40 | 39 | itgeq2dv 25817 |
. . . . . . 7
⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥) |
| 41 | 5 | imcld 15234 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ) |
| 42 | 15 | simprd 495 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈
𝐿1) |
| 43 | 10 | imcld 15234 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℝ) |
| 44 | 19 | simprd 495 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈
𝐿1) |
| 45 | | imf 15152 |
. . . . . . . . . . . . 13
⊢
ℑ:ℂ⟶ℝ |
| 46 | 45 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 →
ℑ:ℂ⟶ℝ) |
| 47 | 46 | feqmptd 6977 |
. . . . . . . . . . 11
⊢ (𝜑 → ℑ = (𝑦 ∈ ℂ ↦
(ℑ‘𝑦))) |
| 48 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐵 + 𝐶) → (ℑ‘𝑦) = (ℑ‘(𝐵 + 𝐶))) |
| 49 | 21, 22, 47, 48 | fmptco 7149 |
. . . . . . . . . 10
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 + 𝐶)))) |
| 50 | 39 | mpteq2dva 5242 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶)))) |
| 51 | 49, 50 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶)))) |
| 52 | 34 | simprd 495 |
. . . . . . . . 9
⊢ (𝜑 → (ℑ ∘ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))) ∈ MblFn) |
| 53 | 51, 52 | eqeltrrd 2842 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐵) + (ℑ‘𝐶))) ∈ MblFn) |
| 54 | 41, 42, 43, 44, 53, 41, 43 | itgaddnclem2 37686 |
. . . . . . 7
⊢ (𝜑 → ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) |
| 55 | 40, 54 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) |
| 56 | 55 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥))) |
| 57 | | ax-icn 11214 |
. . . . . . 7
⊢ i ∈
ℂ |
| 58 | 57 | a1i 11 |
. . . . . 6
⊢ (𝜑 → i ∈
ℂ) |
| 59 | 41, 42 | itgcl 25819 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) |
| 60 | 43, 44 | itgcl 25819 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) |
| 61 | 58, 59, 60 | adddid 11285 |
. . . . 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 2777 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
| 63 | 38, 62 | oveq12d 7449 |
. . 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 25819 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ) |
| 65 | 17, 20 | itgcl 25819 |
. . . 4
⊢ (𝜑 → ∫𝐴(ℜ‘𝐶) d𝑥 ∈ ℂ) |
| 66 | | mulcl 11239 |
. . . . 5
⊢ ((i
∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i ·
∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ) |
| 67 | 57, 59, 66 | sylancr 587 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ) |
| 68 | | mulcl 11239 |
. . . . 5
⊢ ((i
∈ ℂ ∧ ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) → (i ·
∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ) |
| 69 | 57, 60, 68 | sylancr 587 |
. . . 4
⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ) |
| 70 | 64, 65, 67, 69 | add4d 11490 |
. . 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 2777 |
. 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 7466 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ V) |
| 73 | 4, 1, 9, 6, 30 | ibladdnc 37684 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈
𝐿1) |
| 74 | 72, 73 | itgcnval 25835 |
. 2
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥))) |
| 75 | 4, 1 | itgcnval 25835 |
. . 3
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥))) |
| 76 | 9, 6 | itgcnval 25835 |
. . 3
⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) |
| 77 | 75, 76 | oveq12d 7449 |
. 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 2787 |
1
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) |