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 25672 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ dom vol) |
| 4 | | mbfmulc2.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 5 | 4 | recld 15233 |
. . . . . . . 8
⊢ (𝜑 → (ℜ‘𝐶) ∈
ℝ) |
| 6 | 5 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℝ) |
| 7 | 6 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℂ) |
| 8 | 1, 2 | mbfmptcl 25671 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
| 9 | 8 | recld 15233 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ) |
| 10 | 9 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ) |
| 11 | 7, 10 | mulcld 11281 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℜ‘𝐶) · (ℜ‘𝐵)) ∈ ℂ) |
| 12 | | ovexd 7466 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-(ℑ‘𝐶) · (ℑ‘𝐵)) ∈ V) |
| 13 | | fconstmpt 5747 |
. . . . . . 7
⊢ (𝐴 × {(ℜ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) |
| 14 | 13 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴 × {(ℜ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶))) |
| 15 | | eqidd 2738 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) |
| 16 | 3, 6, 9, 14, 15 | offval2 7717 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐶) · (ℜ‘𝐵)))) |
| 17 | 4 | imcld 15234 |
. . . . . . . 8
⊢ (𝜑 → (ℑ‘𝐶) ∈
ℝ) |
| 18 | 17 | renegcld 11690 |
. . . . . . 7
⊢ (𝜑 → -(ℑ‘𝐶) ∈
ℝ) |
| 19 | 18 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐶) ∈ ℝ) |
| 20 | 8 | imcld 15234 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ) |
| 21 | | fconstmpt 5747 |
. . . . . . 7
⊢ (𝐴 × {-(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ -(ℑ‘𝐶)) |
| 22 | 21 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴 × {-(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ -(ℑ‘𝐶))) |
| 23 | | eqidd 2738 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) |
| 24 | 3, 19, 20, 22, 23 | offval2 7717 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {-(ℑ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ (-(ℑ‘𝐶) · (ℑ‘𝐵)))) |
| 25 | 3, 11, 12, 16, 24 | offval2 7717 |
. . . 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))))) |
| 26 | 17 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℝ) |
| 27 | 26 | recnd 11289 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℂ) |
| 28 | 20 | recnd 11289 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ) |
| 29 | 27, 28 | mulcld 11281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℑ‘𝐶) · (ℑ‘𝐵)) ∈ ℂ) |
| 30 | 11, 29 | negsubd 11626 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℜ‘𝐶) · (ℜ‘𝐵)) + -((ℑ‘𝐶) · (ℑ‘𝐵))) = (((ℜ‘𝐶) · (ℜ‘𝐵)) − ((ℑ‘𝐶) · (ℑ‘𝐵)))) |
| 31 | 27, 28 | mulneg1d 11716 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-(ℑ‘𝐶) · (ℑ‘𝐵)) = -((ℑ‘𝐶) · (ℑ‘𝐵))) |
| 32 | 31 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵))) = (((ℜ‘𝐶) · (ℜ‘𝐵)) + -((ℑ‘𝐶) · (ℑ‘𝐵)))) |
| 33 | 4 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 34 | 33, 8 | remuld 15257 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 · 𝐵)) = (((ℜ‘𝐶) · (ℜ‘𝐵)) − ((ℑ‘𝐶) · (ℑ‘𝐵)))) |
| 35 | 30, 32, 34 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵))) = (ℜ‘(𝐶 · 𝐵))) |
| 36 | 35 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (((ℜ‘𝐶) · (ℜ‘𝐵)) + (-(ℑ‘𝐶) · (ℑ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵)))) |
| 37 | 25, 36 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∘f + ((𝐴 × {-(ℑ‘𝐶)}) ∘f ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵)))) |
| 38 | 8 | ismbfcn2 25673 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) |
| 39 | 1, 38 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn)) |
| 40 | 39 | simpld 494 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn) |
| 41 | 10 | fmpttd 7135 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)):𝐴⟶ℂ) |
| 42 | 40, 5, 41 | mbfmulc2re 25683 |
. . . 4
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∈ MblFn) |
| 43 | 39 | simprd 495 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn) |
| 44 | 28 | fmpttd 7135 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)):𝐴⟶ℂ) |
| 45 | 43, 18, 44 | mbfmulc2re 25683 |
. . . 4
⊢ (𝜑 → ((𝐴 × {-(ℑ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∈ MblFn) |
| 46 | 42, 45 | mbfadd 25696 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∘f + ((𝐴 × {-(ℑ‘𝐶)}) ∘f ·
(𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)))) ∈ MblFn) |
| 47 | 37, 46 | eqeltrrd 2842 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵))) ∈ MblFn) |
| 48 | | ovexd 7466 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℜ‘𝐶) · (ℑ‘𝐵)) ∈ V) |
| 49 | | ovexd 7466 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((ℑ‘𝐶) · (ℜ‘𝐵)) ∈ V) |
| 50 | 3, 6, 20, 14, 23 | offval2 7717 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ ((ℜ‘𝐶) · (ℑ‘𝐵)))) |
| 51 | | fconstmpt 5747 |
. . . . . . 7
⊢ (𝐴 × {(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) |
| 52 | 51 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴 × {(ℑ‘𝐶)}) = (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶))) |
| 53 | 3, 26, 9, 52, 15 | offval2 7717 |
. . . . 5
⊢ (𝜑 → ((𝐴 × {(ℑ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) = (𝑥 ∈ 𝐴 ↦ ((ℑ‘𝐶) · (ℜ‘𝐵)))) |
| 54 | 3, 48, 49, 50, 53 | offval2 7717 |
. . . 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))))) |
| 55 | 33, 8 | immuld 15258 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐶 · 𝐵)) = (((ℜ‘𝐶) · (ℑ‘𝐵)) + ((ℑ‘𝐶) · (ℜ‘𝐵)))) |
| 56 | 55 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵))) = (𝑥 ∈ 𝐴 ↦ (((ℜ‘𝐶) · (ℑ‘𝐵)) + ((ℑ‘𝐶) · (ℜ‘𝐵))))) |
| 57 | 54, 56 | eqtr4d 2780 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∘f + ((𝐴 × {(ℑ‘𝐶)}) ∘f ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)))) = (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵)))) |
| 58 | 43, 5, 44 | mbfmulc2re 25683 |
. . . 4
⊢ (𝜑 → ((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∈ MblFn) |
| 59 | 40, 17, 41 | mbfmulc2re 25683 |
. . . 4
⊢ (𝜑 → ((𝐴 × {(ℑ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵))) ∈ MblFn) |
| 60 | 58, 59 | mbfadd 25696 |
. . 3
⊢ (𝜑 → (((𝐴 × {(ℜ‘𝐶)}) ∘f · (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵))) ∘f + ((𝐴 × {(ℑ‘𝐶)}) ∘f ·
(𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)))) ∈ MblFn) |
| 61 | 57, 60 | eqeltrrd 2842 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵))) ∈ MblFn) |
| 62 | 33, 8 | mulcld 11281 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ) |
| 63 | 62 | ismbfcn2 25673 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘(𝐶 · 𝐵))) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘(𝐶 · 𝐵))) ∈ MblFn))) |
| 64 | 47, 61, 63 | mpbir2and 713 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn) |