Step | Hyp | Ref
| Expression |
1 | | mbfmul.1 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ MblFn) |
2 | | mbff 23200 |
. . . . 5
⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ) |
4 | | ffn 5958 |
. . . 4
⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹) |
5 | 3, 4 | syl 17 |
. . 3
⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
6 | | mbfmul.2 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ MblFn) |
7 | | mbff 23200 |
. . . . 5
⊢ (𝐺 ∈ MblFn → 𝐺:dom 𝐺⟶ℂ) |
8 | 6, 7 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐺:dom 𝐺⟶ℂ) |
9 | | ffn 5958 |
. . . 4
⊢ (𝐺:dom 𝐺⟶ℂ → 𝐺 Fn dom 𝐺) |
10 | 8, 9 | syl 17 |
. . 3
⊢ (𝜑 → 𝐺 Fn dom 𝐺) |
11 | | mbfdm 23201 |
. . . 4
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom
vol) |
12 | 1, 11 | syl 17 |
. . 3
⊢ (𝜑 → dom 𝐹 ∈ dom vol) |
13 | | mbfdm 23201 |
. . . 4
⊢ (𝐺 ∈ MblFn → dom 𝐺 ∈ dom
vol) |
14 | 6, 13 | syl 17 |
. . 3
⊢ (𝜑 → dom 𝐺 ∈ dom vol) |
15 | | eqid 2610 |
. . 3
⊢ (dom
𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺) |
16 | | eqidd 2611 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥)) |
17 | | eqidd 2611 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) = (𝐺‘𝑥)) |
18 | 5, 10, 12, 14, 15, 16, 17 | offval 6802 |
. 2
⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))) |
19 | | elin 3758 |
. . . . . . . . 9
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↔ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺)) |
20 | 19 | simplbi 475 |
. . . . . . . 8
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑥 ∈ dom 𝐹) |
21 | | ffvelrn 6265 |
. . . . . . . 8
⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ℂ) |
22 | 3, 20, 21 | syl2an 493 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐹‘𝑥) ∈ ℂ) |
23 | 19 | simprbi 479 |
. . . . . . . 8
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑥 ∈ dom 𝐺) |
24 | | ffvelrn 6265 |
. . . . . . . 8
⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ ℂ) |
25 | 8, 23, 24 | syl2an 493 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐺‘𝑥) ∈ ℂ) |
26 | 22, 25 | remuld 13806 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (ℜ‘((𝐹‘𝑥) · (𝐺‘𝑥))) = (((ℜ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) − ((ℑ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))))) |
27 | 26 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((ℜ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) − ((ℑ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥)))))) |
28 | | inmbl 23117 |
. . . . . . 7
⊢ ((dom
𝐹 ∈ dom vol ∧ dom
𝐺 ∈ dom vol) →
(dom 𝐹 ∩ dom 𝐺) ∈ dom
vol) |
29 | 12, 14, 28 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) |
30 | | ovex 6577 |
. . . . . . 7
⊢
((ℜ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) ∈ V |
31 | 30 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((ℜ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) ∈ V) |
32 | | ovex 6577 |
. . . . . . 7
⊢
((ℑ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) ∈ V |
33 | 32 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((ℑ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) ∈ V) |
34 | 22 | recld 13782 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (ℜ‘(𝐹‘𝑥)) ∈ ℝ) |
35 | 25 | recld 13782 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (ℜ‘(𝐺‘𝑥)) ∈ ℝ) |
36 | | eqidd 2611 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥)))) |
37 | | eqidd 2611 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) |
38 | 29, 34, 35, 36, 37 | offval2 6812 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((ℜ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))))) |
39 | 22 | imcld 13783 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (ℑ‘(𝐹‘𝑥)) ∈ ℝ) |
40 | 25 | imcld 13783 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (ℑ‘(𝐺‘𝑥)) ∈ ℝ) |
41 | | eqidd 2611 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥)))) |
42 | | eqidd 2611 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) |
43 | 29, 39, 40, 41, 42 | offval2 6812 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((ℑ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))))) |
44 | 29, 31, 33, 38, 43 | offval2 6812 |
. . . . 5
⊢ (𝜑 → (((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) ∘𝑓 −
((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((ℜ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) − ((ℑ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥)))))) |
45 | 27, 44 | eqtr4d 2647 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) = (((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) ∘𝑓 −
((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))))) |
46 | | inss1 3795 |
. . . . . . . . . 10
⊢ (dom
𝐹 ∩ dom 𝐺) ⊆ dom 𝐹 |
47 | | resmpt 5369 |
. . . . . . . . . 10
⊢ ((dom
𝐹 ∩ dom 𝐺) ⊆ dom 𝐹 → ((𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹‘𝑥))) |
48 | 46, 47 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹‘𝑥)) |
49 | 3 | feqmptd 6159 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥))) |
50 | 49, 1 | eqeltrrd 2689 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) ∈ MblFn) |
51 | | mbfres 23217 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) ∈ MblFn ∧ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) → ((𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn) |
52 | 50, 29, 51 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ↦ (𝐹‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn) |
53 | 48, 52 | syl5eqelr 2693 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹‘𝑥)) ∈ MblFn) |
54 | 22 | ismbfcn2 23212 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹‘𝑥)) ∈ MblFn ↔ ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn))) |
55 | 53, 54 | mpbid 221 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn)) |
56 | 55 | simpld 474 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∈ MblFn) |
57 | | inss2 3796 |
. . . . . . . . . 10
⊢ (dom
𝐹 ∩ dom 𝐺) ⊆ dom 𝐺 |
58 | | resmpt 5369 |
. . . . . . . . . 10
⊢ ((dom
𝐹 ∩ dom 𝐺) ⊆ dom 𝐺 → ((𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺‘𝑥))) |
59 | 57, 58 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺‘𝑥)) |
60 | 8 | feqmptd 6159 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 = (𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥))) |
61 | 60, 6 | eqeltrrd 2689 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥)) ∈ MblFn) |
62 | | mbfres 23217 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥)) ∈ MblFn ∧ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) → ((𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn) |
63 | 61, 29, 62 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ dom 𝐺 ↦ (𝐺‘𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn) |
64 | 59, 63 | syl5eqelr 2693 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺‘𝑥)) ∈ MblFn) |
65 | 25 | ismbfcn2 23212 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺‘𝑥)) ∈ MblFn ↔ ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))) ∈ MblFn))) |
66 | 64, 65 | mpbid 221 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))) ∈ MblFn ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))) ∈ MblFn)) |
67 | 66 | simpld 474 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))) ∈ MblFn) |
68 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) |
69 | 34, 68 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))):(dom 𝐹 ∩ dom 𝐺)⟶ℝ) |
70 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))) |
71 | 35, 70 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))):(dom 𝐹 ∩ dom 𝐺)⟶ℝ) |
72 | 56, 67, 69, 71 | mbfmullem 23298 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) ∈ MblFn) |
73 | 55 | simprd 478 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∈ MblFn) |
74 | 66 | simprd 478 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))) ∈ MblFn) |
75 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) |
76 | 39, 75 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))):(dom 𝐹 ∩ dom 𝐺)⟶ℝ) |
77 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))) |
78 | 40, 77 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))):(dom 𝐹 ∩ dom 𝐺)⟶ℝ) |
79 | 73, 74, 76, 78 | mbfmullem 23298 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) ∈ MblFn) |
80 | 72, 79 | mbfsub 23235 |
. . . 4
⊢ (𝜑 → (((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) ∘𝑓 −
((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥))))) ∈ MblFn) |
81 | 45, 80 | eqeltrd 2688 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) ∈ MblFn) |
82 | 22, 25 | immuld 13807 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (ℑ‘((𝐹‘𝑥) · (𝐺‘𝑥))) = (((ℜ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) + ((ℑ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))))) |
83 | 82 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((ℜ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) + ((ℑ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥)))))) |
84 | | ovex 6577 |
. . . . . . 7
⊢
((ℜ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) ∈ V |
85 | 84 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((ℜ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) ∈ V) |
86 | | ovex 6577 |
. . . . . . 7
⊢
((ℑ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) ∈ V |
87 | 86 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((ℑ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))) ∈ V) |
88 | 29, 34, 40, 36, 42 | offval2 6812 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((ℜ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))))) |
89 | 29, 39, 35, 41, 37 | offval2 6812 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((ℑ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥))))) |
90 | 29, 85, 87, 88, 89 | offval2 6812 |
. . . . 5
⊢ (𝜑 → (((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) ∘𝑓 + ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((ℜ‘(𝐹‘𝑥)) · (ℑ‘(𝐺‘𝑥))) + ((ℑ‘(𝐹‘𝑥)) · (ℜ‘(𝐺‘𝑥)))))) |
91 | 83, 90 | eqtr4d 2647 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) = (((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) ∘𝑓 + ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))))) |
92 | 56, 74, 69, 78 | mbfmullem 23298 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) ∈ MblFn) |
93 | 73, 67, 76, 71 | mbfmullem 23298 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥)))) ∈ MblFn) |
94 | 92, 93 | mbfadd 23234 |
. . . 4
⊢ (𝜑 → (((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐺‘𝑥)))) ∘𝑓 + ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘(𝐹‘𝑥))) ∘𝑓 ·
(𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘(𝐺‘𝑥))))) ∈ MblFn) |
95 | 91, 94 | eqeltrd 2688 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) ∈ MblFn) |
96 | 22, 25 | mulcld 9939 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹‘𝑥) · (𝐺‘𝑥)) ∈ ℂ) |
97 | 96 | ismbfcn2 23212 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥))) ∈ MblFn ↔ ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℜ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) ∈ MblFn ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (ℑ‘((𝐹‘𝑥) · (𝐺‘𝑥)))) ∈ MblFn))) |
98 | 81, 95, 97 | mpbir2and 959 |
. 2
⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥))) ∈ MblFn) |
99 | 18, 98 | eqeltrd 2688 |
1
⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) ∈ MblFn) |