Step | Hyp | Ref
| Expression |
1 | | dvaddf.f |
. . . . 5
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
2 | 1 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶ℂ) |
3 | | dvaddf.df |
. . . . . 6
⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
4 | | dvbsss 25066 |
. . . . . 6
⊢ dom
(𝑆 D 𝐹) ⊆ 𝑆 |
5 | 3, 4 | eqsstrrdi 3976 |
. . . . 5
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | 5 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑋 ⊆ 𝑆) |
7 | | dvaddf.g |
. . . . 5
⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
8 | 7 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺:𝑋⟶ℂ) |
9 | | dvaddf.s |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
10 | 9 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ {ℝ, ℂ}) |
11 | 3 | eleq2d 2824 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥 ∈ 𝑋)) |
12 | 11 | biimpar 478 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D 𝐹)) |
13 | | dvaddf.dg |
. . . . . 6
⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
14 | 13 | eleq2d 2824 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥 ∈ 𝑋)) |
15 | 14 | biimpar 478 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D 𝐺)) |
16 | 2, 6, 8, 6, 10, 12, 15 | dvmul 25105 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D (𝐹 ∘f · 𝐺))‘𝑥) = ((((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)) + (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥)))) |
17 | 16 | mpteq2dva 5174 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝑆 D (𝐹 ∘f · 𝐺))‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ((((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)) + (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥))))) |
18 | | dvfg 25070 |
. . . . 5
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D (𝐹 ∘f ·
𝐺)):dom (𝑆 D (𝐹 ∘f · 𝐺))⟶ℂ) |
19 | 9, 18 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)):dom (𝑆 D (𝐹 ∘f · 𝐺))⟶ℂ) |
20 | | recnprss 25068 |
. . . . . . . 8
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
21 | 9, 20 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
22 | | mulcl 10955 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
23 | 22 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ) |
24 | 9, 5 | ssexd 5248 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ V) |
25 | | inidm 4152 |
. . . . . . . 8
⊢ (𝑋 ∩ 𝑋) = 𝑋 |
26 | 23, 1, 7, 24, 24, 25 | off 7551 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ∘f · 𝐺):𝑋⟶ℂ) |
27 | 21, 26, 5 | dvbss 25065 |
. . . . . 6
⊢ (𝜑 → dom (𝑆 D (𝐹 ∘f · 𝐺)) ⊆ 𝑋) |
28 | 21 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ ℂ) |
29 | | fvexd 6789 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐹)‘𝑥) ∈ V) |
30 | | fvexd 6789 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑆 D 𝐺)‘𝑥) ∈ V) |
31 | | dvfg 25070 |
. . . . . . . . . . . 12
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
32 | 9, 31 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
33 | 32 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
34 | | ffun 6603 |
. . . . . . . . . 10
⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) |
35 | | funfvbrb 6928 |
. . . . . . . . . 10
⊢ (Fun
(𝑆 D 𝐹) → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥))) |
36 | 33, 34, 35 | 3syl 18 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ dom (𝑆 D 𝐹) ↔ 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥))) |
37 | 12, 36 | mpbid 231 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝑥)) |
38 | | dvfg 25070 |
. . . . . . . . . . . 12
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
39 | 9, 38 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
40 | 39 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
41 | | ffun 6603 |
. . . . . . . . . 10
⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺)) |
42 | | funfvbrb 6928 |
. . . . . . . . . 10
⊢ (Fun
(𝑆 D 𝐺) → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥))) |
43 | 40, 41, 42 | 3syl 18 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ dom (𝑆 D 𝐺) ↔ 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥))) |
44 | 15, 43 | mpbid 231 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝑥)) |
45 | | eqid 2738 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
46 | 2, 6, 8, 6, 28, 29, 30, 37, 44, 45 | dvmulbr 25103 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥(𝑆 D (𝐹 ∘f · 𝐺))((((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)) + (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥)))) |
47 | | reldv 25034 |
. . . . . . . 8
⊢ Rel
(𝑆 D (𝐹 ∘f · 𝐺)) |
48 | 47 | releldmi 5857 |
. . . . . . 7
⊢ (𝑥(𝑆 D (𝐹 ∘f · 𝐺))((((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)) + (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥))) → 𝑥 ∈ dom (𝑆 D (𝐹 ∘f · 𝐺))) |
49 | 46, 48 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom (𝑆 D (𝐹 ∘f · 𝐺))) |
50 | 27, 49 | eqelssd 3942 |
. . . . 5
⊢ (𝜑 → dom (𝑆 D (𝐹 ∘f · 𝐺)) = 𝑋) |
51 | 50 | feq2d 6586 |
. . . 4
⊢ (𝜑 → ((𝑆 D (𝐹 ∘f · 𝐺)):dom (𝑆 D (𝐹 ∘f · 𝐺))⟶ℂ ↔ (𝑆 D (𝐹 ∘f · 𝐺)):𝑋⟶ℂ)) |
52 | 19, 51 | mpbid 231 |
. . 3
⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)):𝑋⟶ℂ) |
53 | 52 | feqmptd 6837 |
. 2
⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D (𝐹 ∘f · 𝐺))‘𝑥))) |
54 | | ovexd 7310 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)) ∈ V) |
55 | | ovexd 7310 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥)) ∈ V) |
56 | | fvexd 6789 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ V) |
57 | 3 | feq2d 6586 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
58 | 32, 57 | mpbid 231 |
. . . . 5
⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
59 | 58 | feqmptd 6837 |
. . . 4
⊢ (𝜑 → (𝑆 D 𝐹) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑥))) |
60 | 7 | feqmptd 6837 |
. . . 4
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ (𝐺‘𝑥))) |
61 | 24, 29, 56, 59, 60 | offval2 7553 |
. . 3
⊢ (𝜑 → ((𝑆 D 𝐹) ∘f · 𝐺) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)))) |
62 | | fvexd 6789 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ V) |
63 | 13 | feq2d 6586 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ ↔ (𝑆 D 𝐺):𝑋⟶ℂ)) |
64 | 39, 63 | mpbid 231 |
. . . . 5
⊢ (𝜑 → (𝑆 D 𝐺):𝑋⟶ℂ) |
65 | 64 | feqmptd 6837 |
. . . 4
⊢ (𝜑 → (𝑆 D 𝐺) = (𝑥 ∈ 𝑋 ↦ ((𝑆 D 𝐺)‘𝑥))) |
66 | 1 | feqmptd 6837 |
. . . 4
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥))) |
67 | 24, 30, 62, 65, 66 | offval2 7553 |
. . 3
⊢ (𝜑 → ((𝑆 D 𝐺) ∘f · 𝐹) = (𝑥 ∈ 𝑋 ↦ (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥)))) |
68 | 24, 54, 55, 61, 67 | offval2 7553 |
. 2
⊢ (𝜑 → (((𝑆 D 𝐹) ∘f · 𝐺) ∘f + ((𝑆 D 𝐺) ∘f · 𝐹)) = (𝑥 ∈ 𝑋 ↦ ((((𝑆 D 𝐹)‘𝑥) · (𝐺‘𝑥)) + (((𝑆 D 𝐺)‘𝑥) · (𝐹‘𝑥))))) |
69 | 17, 53, 68 | 3eqtr4d 2788 |
1
⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) = (((𝑆 D 𝐹) ∘f · 𝐺) ∘f + ((𝑆 D 𝐺) ∘f · 𝐹))) |