Proof of Theorem mbfmulc2
Step | Hyp | Ref
| Expression |
1 | | mbfmulc2.3 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
2 | | mbfmulc2.2 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
3 | 1, 2 | mbfdm2 23211 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ dom vol) |
4 | | mbfmulc2.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℂ) |
5 | 4 | recld 13782 |
. . . . . . . 8
⊢ (𝜑 → (ℜ‘𝐶) ∈
ℝ) |
6 | 5 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℝ) |
7 | 6 | recnd 9947 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℂ) |
8 | 1, 2 | mbfmptcl 23210 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
9 | 8 | recld 13782 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ) |
10 | 9 | recnd 9947 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ) |
11 | 7, 10 | mulcld 9939 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℜ‘𝐶) · (ℜ‘𝐵)) ∈ ℂ) |
12 | | ovex 6577 |
. . . . . 6
⊢
(-(ℑ‘𝐶)
· (ℑ‘𝐵))
∈ V |
13 | 12 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-(ℑ‘𝐶) · (ℑ‘𝐵)) ∈ V) |
14 | | fconstmpt 5085 |
. . . . . . 7
⊢ (𝐴 × {(ℜ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) |
15 | 14 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴 × {(ℜ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶))) |
16 | | eqidd 2611 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) |
17 | 3, 6, 9, 15, 16 | offval2 6812 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐶) · (ℜ‘𝐵)))) |
18 | 4 | imcld 13783 |
. . . . . . . 8
⊢ (𝜑 → (ℑ‘𝐶) ∈
ℝ) |
19 | 18 | renegcld 10336 |
. . . . . . 7
⊢ (𝜑 → -(ℑ‘𝐶) ∈
ℝ) |
20 | 19 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐶) ∈ ℝ) |
21 | 8 | imcld 13783 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ) |
22 | | fconstmpt 5085 |
. . . . . . 7
⊢ (𝐴 × {-(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ -(ℑ‘𝐶)) |
23 | 22 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴 × {-(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ -(ℑ‘𝐶))) |
24 | | eqidd 2611 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) |
25 | 3, 20, 21, 23, 24 | offval2 6812 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {-(ℑ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ (-(ℑ‘𝐶) · (ℑ‘𝐵)))) |
26 | 3, 11, 13, 17, 25 | offval2 6812 |
. . . 4
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∘𝑓 + ((𝐴 × {-(ℑ‘𝐶)}) ∘𝑓
· (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵))))) |
27 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℝ) |
28 | 27 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℂ) |
29 | 21 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ) |
30 | 28, 29 | mulcld 9939 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℑ‘𝐶) · (ℑ‘𝐵)) ∈ ℂ) |
31 | 11, 30 | negsubd 10277 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℜ‘𝐶) · (ℜ‘𝐵)) + -((ℑ‘𝐶) · (ℑ‘𝐵))) = (((ℜ‘𝐶) · (ℜ‘𝐵)) − ((ℑ‘𝐶) · (ℑ‘𝐵)))) |
32 | 28, 29 | mulneg1d 10362 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-(ℑ‘𝐶) · (ℑ‘𝐵)) = -((ℑ‘𝐶) · (ℑ‘𝐵))) |
33 | 32 | oveq2d 6565 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵))) = (((ℜ‘𝐶) · (ℜ‘𝐵)) + -((ℑ‘𝐶) · (ℑ‘𝐵)))) |
34 | 4 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
35 | 34, 8 | remuld 13806 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 · 𝐵)) = (((ℜ‘𝐶) · (ℜ‘𝐵)) − ((ℑ‘𝐶) · (ℑ‘𝐵)))) |
36 | 31, 33, 35 | 3eqtr4d 2654 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵))) = (ℜ‘(𝐶 · 𝐵))) |
37 | 36 | mpteq2dva 4672 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵)))) |
38 | 26, 37 | eqtrd 2644 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∘𝑓 + ((𝐴 × {-(ℑ‘𝐶)}) ∘𝑓
· (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵)))) |
39 | 8 | ismbfcn2 23212 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) |
40 | 1, 39 | mpbid 221 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn)) |
41 | 40 | simpld 474 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn) |
42 | | eqid 2610 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) |
43 | 10, 42 | fmptd 6292 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)):𝐴⟶ℂ) |
44 | 41, 5, 43 | mbfmulc2re 23221 |
. . . 4
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∈ MblFn) |
45 | 40 | simprd 478 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn) |
46 | | eqid 2610 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) |
47 | 29, 46 | fmptd 6292 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)):𝐴⟶ℂ) |
48 | 45, 19, 47 | mbfmulc2re 23221 |
. . . 4
⊢ (𝜑 → ((𝐴 × {-(ℑ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∈ MblFn) |
49 | 44, 48 | mbfadd 23234 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∘𝑓 + ((𝐴 × {-(ℑ‘𝐶)}) ∘𝑓
· (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)))) ∈
MblFn) |
50 | 38, 49 | eqeltrrd 2689 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵))) ∈ MblFn) |
51 | | ovex 6577 |
. . . . . 6
⊢
((ℜ‘𝐶)
· (ℑ‘𝐵))
∈ V |
52 | 51 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℜ‘𝐶) · (ℑ‘𝐵)) ∈ V) |
53 | | ovex 6577 |
. . . . . 6
⊢
((ℑ‘𝐶)
· (ℜ‘𝐵))
∈ V |
54 | 53 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℑ‘𝐶) · (ℜ‘𝐵)) ∈ V) |
55 | 3, 6, 21, 15, 24 | offval2 6812 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐶) · (ℑ‘𝐵)))) |
56 | | fconstmpt 5085 |
. . . . . . 7
⊢ (𝐴 × {(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) |
57 | 56 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴 × {(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶))) |
58 | 3, 27, 9, 57, 16 | offval2 6812 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {(ℑ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐶) · (ℜ‘𝐵)))) |
59 | 3, 52, 54, 55, 58 | offval2 6812 |
. . . 4
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∘𝑓 + ((𝐴 × {(ℑ‘𝐶)}) ∘𝑓
· (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (((ℜ‘𝐶) · (ℑ‘𝐵)) + ((ℑ‘𝐶) · (ℜ‘𝐵))))) |
60 | 34, 8 | immuld 13807 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐶 · 𝐵)) = (((ℜ‘𝐶) · (ℑ‘𝐵)) + ((ℑ‘𝐶) · (ℜ‘𝐵)))) |
61 | 60 | mpteq2dva 4672 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵))) = (𝑥 ∈ 𝐴 ↦ (((ℜ‘𝐶) · (ℑ‘𝐵)) + ((ℑ‘𝐶) · (ℜ‘𝐵))))) |
62 | 59, 61 | eqtr4d 2647 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∘𝑓 + ((𝐴 × {(ℑ‘𝐶)}) ∘𝑓
· (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵)))) |
63 | 45, 5, 47 | mbfmulc2re 23221 |
. . . 4
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∈ MblFn) |
64 | 41, 18, 43 | mbfmulc2re 23221 |
. . . 4
⊢ (𝜑 → ((𝐴 × {(ℑ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∈ MblFn) |
65 | 63, 64 | mbfadd 23234 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘𝑓 ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∘𝑓 + ((𝐴 × {(ℑ‘𝐶)}) ∘𝑓
· (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)))) ∈
MblFn) |
66 | 62, 65 | eqeltrrd 2689 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵))) ∈ MblFn) |
67 | 34, 8 | mulcld 9939 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ) |
68 | 67 | ismbfcn2 23212 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵))) ∈ MblFn))) |
69 | 50, 66, 68 | mpbir2and 959 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) |