Step | Hyp | Ref
| Expression |
1 | | dvcmul.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
2 | | dvcmul.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | | fconstg 6559 |
. . . . 5
⊢ (𝐴 ∈ ℂ → (𝑋 × {𝐴}):𝑋⟶{𝐴}) |
4 | 2, 3 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑋 × {𝐴}):𝑋⟶{𝐴}) |
5 | 2 | snssd 4694 |
. . . 4
⊢ (𝜑 → {𝐴} ⊆ ℂ) |
6 | 4, 5 | fssd 6516 |
. . 3
⊢ (𝜑 → (𝑋 × {𝐴}):𝑋⟶ℂ) |
7 | | dvcmul.f |
. . 3
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
8 | | c0ex 10706 |
. . . . . 6
⊢ 0 ∈
V |
9 | 8 | fconst 6558 |
. . . . 5
⊢ (𝑋 × {0}):𝑋⟶{0} |
10 | | recnprss 24648 |
. . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
11 | 1, 10 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
12 | | fconstg 6559 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℂ → (𝑆 × {𝐴}):𝑆⟶{𝐴}) |
13 | 2, 12 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 × {𝐴}):𝑆⟶{𝐴}) |
14 | 13, 5 | fssd 6516 |
. . . . . . . 8
⊢ (𝜑 → (𝑆 × {𝐴}):𝑆⟶ℂ) |
15 | | ssidd 3898 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ⊆ 𝑆) |
16 | | dvcmulf.df |
. . . . . . . . 9
⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
17 | | dvbsss 24646 |
. . . . . . . . . 10
⊢ dom
(𝑆 D 𝐹) ⊆ 𝑆 |
18 | 17 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑆) |
19 | 16, 18 | eqsstrrd 3914 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
20 | | eqid 2738 |
. . . . . . . . 9
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
21 | | eqid 2738 |
. . . . . . . . 9
⊢
((TopOpen‘ℂfld) ↾t 𝑆) =
((TopOpen‘ℂfld) ↾t 𝑆) |
22 | 20, 21 | dvres 24655 |
. . . . . . . 8
⊢ (((𝑆 ⊆ ℂ ∧ (𝑆 × {𝐴}):𝑆⟶ℂ) ∧ (𝑆 ⊆ 𝑆 ∧ 𝑋 ⊆ 𝑆)) → (𝑆 D ((𝑆 × {𝐴}) ↾ 𝑋)) = ((𝑆 D (𝑆 × {𝐴})) ↾
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))) |
23 | 11, 14, 15, 19, 22 | syl22anc 838 |
. . . . . . 7
⊢ (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ↾ 𝑋)) = ((𝑆 D (𝑆 × {𝐴})) ↾
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))) |
24 | 19 | resmptd 5876 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝑆 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴)) |
25 | | fconstmpt 5579 |
. . . . . . . . . 10
⊢ (𝑆 × {𝐴}) = (𝑥 ∈ 𝑆 ↦ 𝐴) |
26 | 25 | reseq1i 5815 |
. . . . . . . . 9
⊢ ((𝑆 × {𝐴}) ↾ 𝑋) = ((𝑥 ∈ 𝑆 ↦ 𝐴) ↾ 𝑋) |
27 | | fconstmpt 5579 |
. . . . . . . . 9
⊢ (𝑋 × {𝐴}) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
28 | 24, 26, 27 | 3eqtr4g 2798 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 × {𝐴}) ↾ 𝑋) = (𝑋 × {𝐴})) |
29 | 28 | oveq2d 7180 |
. . . . . . 7
⊢ (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ↾ 𝑋)) = (𝑆 D (𝑋 × {𝐴}))) |
30 | 19 | resmptd 5876 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝑆 ↦ 0) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 0)) |
31 | | fconstg 6559 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ → (ℂ
× {𝐴}):ℂ⟶{𝐴}) |
32 | 2, 31 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (ℂ × {𝐴}):ℂ⟶{𝐴}) |
33 | 32, 5 | fssd 6516 |
. . . . . . . . . . . 12
⊢ (𝜑 → (ℂ × {𝐴}):ℂ⟶ℂ) |
34 | | ssidd 3898 |
. . . . . . . . . . . 12
⊢ (𝜑 → ℂ ⊆
ℂ) |
35 | | dvconst 24661 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ ℂ → (ℂ
D (ℂ × {𝐴})) =
(ℂ × {0})) |
36 | 2, 35 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (ℂ D (ℂ
× {𝐴})) = (ℂ
× {0})) |
37 | 36 | dmeqd 5742 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom (ℂ D (ℂ
× {𝐴})) = dom
(ℂ × {0})) |
38 | 8 | fconst 6558 |
. . . . . . . . . . . . . . 15
⊢ (ℂ
× {0}):ℂ⟶{0} |
39 | 38 | fdmi 6510 |
. . . . . . . . . . . . . 14
⊢ dom
(ℂ × {0}) = ℂ |
40 | 37, 39 | eqtrdi 2789 |
. . . . . . . . . . . . 13
⊢ (𝜑 → dom (ℂ D (ℂ
× {𝐴})) =
ℂ) |
41 | 11, 40 | sseqtrrd 3916 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ⊆ dom (ℂ D (ℂ ×
{𝐴}))) |
42 | | dvres3 24657 |
. . . . . . . . . . . 12
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧
(ℂ × {𝐴}):ℂ⟶ℂ) ∧ (ℂ
⊆ ℂ ∧ 𝑆
⊆ dom (ℂ D (ℂ × {𝐴})))) → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) |
43 | 1, 33, 34, 41, 42 | syl22anc 838 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = ((ℂ D (ℂ × {𝐴})) ↾ 𝑆)) |
44 | | xpssres 5856 |
. . . . . . . . . . . . 13
⊢ (𝑆 ⊆ ℂ →
((ℂ × {𝐴})
↾ 𝑆) = (𝑆 × {𝐴})) |
45 | 11, 44 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((ℂ × {𝐴}) ↾ 𝑆) = (𝑆 × {𝐴})) |
46 | 45 | oveq2d 7180 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 D ((ℂ × {𝐴}) ↾ 𝑆)) = (𝑆 D (𝑆 × {𝐴}))) |
47 | 36 | reseq1d 5818 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((ℂ D (ℂ
× {𝐴})) ↾ 𝑆) = ((ℂ × {0})
↾ 𝑆)) |
48 | | xpssres 5856 |
. . . . . . . . . . . . 13
⊢ (𝑆 ⊆ ℂ →
((ℂ × {0}) ↾ 𝑆) = (𝑆 × {0})) |
49 | 11, 48 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((ℂ × {0})
↾ 𝑆) = (𝑆 × {0})) |
50 | 47, 49 | eqtrd 2773 |
. . . . . . . . . . 11
⊢ (𝜑 → ((ℂ D (ℂ
× {𝐴})) ↾ 𝑆) = (𝑆 × {0})) |
51 | 43, 46, 50 | 3eqtr3d 2781 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0})) |
52 | | fconstmpt 5579 |
. . . . . . . . . 10
⊢ (𝑆 × {0}) = (𝑥 ∈ 𝑆 ↦ 0) |
53 | 51, 52 | eqtrdi 2789 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 D (𝑆 × {𝐴})) = (𝑥 ∈ 𝑆 ↦ 0)) |
54 | 20 | cnfldtopon 23528 |
. . . . . . . . . . . . 13
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
55 | | resttopon 21905 |
. . . . . . . . . . . . 13
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
56 | 54, 11, 55 | sylancr 590 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
57 | | topontop 21657 |
. . . . . . . . . . . 12
⊢
(((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) →
((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top) |
58 | 56, 57 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top) |
59 | | toponuni 21658 |
. . . . . . . . . . . . 13
⊢
(((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪
((TopOpen‘ℂfld) ↾t 𝑆)) |
60 | 56, 59 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 = ∪
((TopOpen‘ℂfld) ↾t 𝑆)) |
61 | 19, 60 | sseqtrd 3915 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ⊆ ∪
((TopOpen‘ℂfld) ↾t 𝑆)) |
62 | | eqid 2738 |
. . . . . . . . . . . 12
⊢ ∪ ((TopOpen‘ℂfld)
↾t 𝑆) =
∪ ((TopOpen‘ℂfld)
↾t 𝑆) |
63 | 62 | ntrss2 21801 |
. . . . . . . . . . 11
⊢
((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ ((TopOpen‘ℂfld)
↾t 𝑆))
→ ((int‘((TopOpen‘ℂfld) ↾t
𝑆))‘𝑋) ⊆ 𝑋) |
64 | 58, 61, 63 | syl2anc 587 |
. . . . . . . . . 10
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ⊆ 𝑋) |
65 | 11, 7, 19, 21, 20 | dvbssntr 24644 |
. . . . . . . . . . 11
⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
66 | 16, 65 | eqsstrrd 3914 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ⊆
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) |
67 | 64, 66 | eqssd 3892 |
. . . . . . . . 9
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋) |
68 | 53, 67 | reseq12d 5820 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴})) ↾
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) = ((𝑥 ∈ 𝑆 ↦ 0) ↾ 𝑋)) |
69 | | fconstmpt 5579 |
. . . . . . . . 9
⊢ (𝑋 × {0}) = (𝑥 ∈ 𝑋 ↦ 0) |
70 | 69 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝑋 × {0}) = (𝑥 ∈ 𝑋 ↦ 0)) |
71 | 30, 68, 70 | 3eqtr4d 2783 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D (𝑆 × {𝐴})) ↾
((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)) = (𝑋 × {0})) |
72 | 23, 29, 71 | 3eqtr3d 2781 |
. . . . . 6
⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0})) |
73 | 72 | feq1d 6483 |
. . . . 5
⊢ (𝜑 → ((𝑆 D (𝑋 × {𝐴})):𝑋⟶{0} ↔ (𝑋 × {0}):𝑋⟶{0})) |
74 | 9, 73 | mpbiri 261 |
. . . 4
⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})):𝑋⟶{0}) |
75 | 74 | fdmd 6509 |
. . 3
⊢ (𝜑 → dom (𝑆 D (𝑋 × {𝐴})) = 𝑋) |
76 | 1, 6, 7, 75, 16 | dvmulf 24687 |
. 2
⊢ (𝜑 → (𝑆 D ((𝑋 × {𝐴}) ∘f · 𝐹)) = (((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) ∘f + ((𝑆 D 𝐹) ∘f · (𝑋 × {𝐴})))) |
77 | | sseqin2 4104 |
. . . . . 6
⊢ (𝑋 ⊆ 𝑆 ↔ (𝑆 ∩ 𝑋) = 𝑋) |
78 | 19, 77 | sylib 221 |
. . . . 5
⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑋) |
79 | 78 | mpteq1d 5116 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑋) ↦ (𝐴 · (𝐹‘𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝐴 · (𝐹‘𝑥)))) |
80 | 13 | ffnd 6499 |
. . . . 5
⊢ (𝜑 → (𝑆 × {𝐴}) Fn 𝑆) |
81 | 7 | ffnd 6499 |
. . . . 5
⊢ (𝜑 → 𝐹 Fn 𝑋) |
82 | 1, 19 | ssexd 5189 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ V) |
83 | | eqid 2738 |
. . . . 5
⊢ (𝑆 ∩ 𝑋) = (𝑆 ∩ 𝑋) |
84 | | fvconst2g 6968 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑆) → ((𝑆 × {𝐴})‘𝑥) = 𝐴) |
85 | 2, 84 | sylan 583 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑆 × {𝐴})‘𝑥) = 𝐴) |
86 | | eqidd 2739 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑥)) |
87 | 80, 81, 1, 82, 83, 85, 86 | offval 7427 |
. . . 4
⊢ (𝜑 → ((𝑆 × {𝐴}) ∘f · 𝐹) = (𝑥 ∈ (𝑆 ∩ 𝑋) ↦ (𝐴 · (𝐹‘𝑥)))) |
88 | 4 | ffnd 6499 |
. . . . 5
⊢ (𝜑 → (𝑋 × {𝐴}) Fn 𝑋) |
89 | | inidm 4107 |
. . . . 5
⊢ (𝑋 ∩ 𝑋) = 𝑋 |
90 | | fvconst2g 6968 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴) |
91 | 2, 90 | sylan 583 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴) |
92 | 88, 81, 82, 82, 89, 91, 86 | offval 7427 |
. . . 4
⊢ (𝜑 → ((𝑋 × {𝐴}) ∘f · 𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐴 · (𝐹‘𝑥)))) |
93 | 79, 87, 92 | 3eqtr4d 2783 |
. . 3
⊢ (𝜑 → ((𝑆 × {𝐴}) ∘f · 𝐹) = ((𝑋 × {𝐴}) ∘f · 𝐹)) |
94 | 93 | oveq2d 7180 |
. 2
⊢ (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹)) = (𝑆 D ((𝑋 × {𝐴}) ∘f · 𝐹))) |
95 | 78 | mpteq1d 5116 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑋) ↦ (𝐴 · ((𝑆 D 𝐹)‘𝑥))) = (𝑥 ∈ 𝑋 ↦ (𝐴 · ((𝑆 D 𝐹)‘𝑥)))) |
96 | | dvfg 24650 |
. . . . . . 7
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
97 | 1, 96 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
98 | 16 | feq2d 6484 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
99 | 97, 98 | mpbid 235 |
. . . . 5
⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
100 | 99 | ffnd 6499 |
. . . 4
⊢ (𝜑 → (𝑆 D 𝐹) Fn 𝑋) |
101 | | eqidd 2739 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) = ((𝑆 D 𝐹)‘𝑥)) |
102 | 80, 100, 1, 82, 83, 85, 101 | offval 7427 |
. . 3
⊢ (𝜑 → ((𝑆 × {𝐴}) ∘f · (𝑆 D 𝐹)) = (𝑥 ∈ (𝑆 ∩ 𝑋) ↦ (𝐴 · ((𝑆 D 𝐹)‘𝑥)))) |
103 | | 0cnd 10705 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℂ) |
104 | | ovexd 7199 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐹)‘𝑥) · 𝐴) ∈ V) |
105 | 72 | oveq1d 7179 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) = ((𝑋 × {0}) ∘f ·
𝐹)) |
106 | | 0cnd 10705 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℂ) |
107 | | mul02 10889 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ → (0
· 𝑥) =
0) |
108 | 107 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
109 | 82, 7, 106, 106, 108 | caofid2 7452 |
. . . . . . 7
⊢ (𝜑 → ((𝑋 × {0}) ∘f ·
𝐹) = (𝑋 × {0})) |
110 | 105, 109 | eqtrd 2773 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) = (𝑋 × {0})) |
111 | 110, 69 | eqtrdi 2789 |
. . . . 5
⊢ (𝜑 → ((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) = (𝑥 ∈ 𝑋 ↦ 0)) |
112 | | fvexd 6683 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) ∈ V) |
113 | 2 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
114 | 99 | feqmptd 6731 |
. . . . . 6
⊢ (𝜑 → (𝑆 D 𝐹) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑥))) |
115 | 27 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑋 × {𝐴}) = (𝑥 ∈ 𝑋 ↦ 𝐴)) |
116 | 82, 112, 113, 114, 115 | offval2 7438 |
. . . . 5
⊢ (𝜑 → ((𝑆 D 𝐹) ∘f · (𝑋 × {𝐴})) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D 𝐹)‘𝑥) · 𝐴))) |
117 | 82, 103, 104, 111, 116 | offval2 7438 |
. . . 4
⊢ (𝜑 → (((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) ∘f + ((𝑆 D 𝐹) ∘f · (𝑋 × {𝐴}))) = (𝑥 ∈ 𝑋 ↦ (0 + (((𝑆 D 𝐹)‘𝑥) · 𝐴)))) |
118 | 99 | ffvelrnda 6855 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) ∈ ℂ) |
119 | 118, 113 | mulcld 10732 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐹)‘𝑥) · 𝐴) ∈ ℂ) |
120 | 119 | addid2d 10912 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (0 + (((𝑆 D 𝐹)‘𝑥) · 𝐴)) = (((𝑆 D 𝐹)‘𝑥) · 𝐴)) |
121 | 118, 113 | mulcomd 10733 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐹)‘𝑥) · 𝐴) = (𝐴 · ((𝑆 D 𝐹)‘𝑥))) |
122 | 120, 121 | eqtrd 2773 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (0 + (((𝑆 D 𝐹)‘𝑥) · 𝐴)) = (𝐴 · ((𝑆 D 𝐹)‘𝑥))) |
123 | 122 | mpteq2dva 5122 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (0 + (((𝑆 D 𝐹)‘𝑥) · 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝐴 · ((𝑆 D 𝐹)‘𝑥)))) |
124 | 117, 123 | eqtrd 2773 |
. . 3
⊢ (𝜑 → (((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) ∘f + ((𝑆 D 𝐹) ∘f · (𝑋 × {𝐴}))) = (𝑥 ∈ 𝑋 ↦ (𝐴 · ((𝑆 D 𝐹)‘𝑥)))) |
125 | 95, 102, 124 | 3eqtr4d 2783 |
. 2
⊢ (𝜑 → ((𝑆 × {𝐴}) ∘f · (𝑆 D 𝐹)) = (((𝑆 D (𝑋 × {𝐴})) ∘f · 𝐹) ∘f + ((𝑆 D 𝐹) ∘f · (𝑋 × {𝐴})))) |
126 | 76, 94, 125 | 3eqtr4d 2783 |
1
⊢ (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹)) = ((𝑆 × {𝐴}) ∘f · (𝑆 D 𝐹))) |