Step | Hyp | Ref
| Expression |
1 | | dgrcolem1.2 |
. 2
⊢ (𝜑 → 𝑀 ∈ ℕ) |
2 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑦 = 1 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑1)) |
3 | 2 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑦 = 1 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) |
4 | 3 | fveq2d 6195 |
. . . . 5
⊢ (𝑦 = 1 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)))) |
5 | | oveq1 6657 |
. . . . 5
⊢ (𝑦 = 1 → (𝑦 · 𝑁) = (1 · 𝑁)) |
6 | 4, 5 | eqeq12d 2637 |
. . . 4
⊢ (𝑦 = 1 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))) |
7 | 6 | imbi2d 330 |
. . 3
⊢ (𝑦 = 1 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)))) |
8 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑦 = 𝑑 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑑)) |
9 | 8 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑦 = 𝑑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
10 | 9 | fveq2d 6195 |
. . . . 5
⊢ (𝑦 = 𝑑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))) |
11 | | oveq1 6657 |
. . . . 5
⊢ (𝑦 = 𝑑 → (𝑦 · 𝑁) = (𝑑 · 𝑁)) |
12 | 10, 11 | eqeq12d 2637 |
. . . 4
⊢ (𝑦 = 𝑑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))) |
13 | 12 | imbi2d 330 |
. . 3
⊢ (𝑦 = 𝑑 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)))) |
14 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑦 = (𝑑 + 1) → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑(𝑑 + 1))) |
15 | 14 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑦 = (𝑑 + 1) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) |
16 | 15 | fveq2d 6195 |
. . . . 5
⊢ (𝑦 = (𝑑 + 1) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))))) |
17 | | oveq1 6657 |
. . . . 5
⊢ (𝑦 = (𝑑 + 1) → (𝑦 · 𝑁) = ((𝑑 + 1) · 𝑁)) |
18 | 16, 17 | eqeq12d 2637 |
. . . 4
⊢ (𝑦 = (𝑑 + 1) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))) |
19 | 18 | imbi2d 330 |
. . 3
⊢ (𝑦 = (𝑑 + 1) → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))) |
20 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑦 = 𝑀 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑀)) |
21 | 20 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑦 = 𝑀 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) |
22 | 21 | fveq2d 6195 |
. . . . 5
⊢ (𝑦 = 𝑀 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) |
23 | | oveq1 6657 |
. . . . 5
⊢ (𝑦 = 𝑀 → (𝑦 · 𝑁) = (𝑀 · 𝑁)) |
24 | 22, 23 | eqeq12d 2637 |
. . . 4
⊢ (𝑦 = 𝑀 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))) |
25 | 24 | imbi2d 330 |
. . 3
⊢ (𝑦 = 𝑀 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)))) |
26 | | dgrcolem1.4 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) |
27 | | plyf 23954 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) |
28 | 26, 27 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺:ℂ⟶ℂ) |
29 | 28 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ) |
30 | 29 | exp1d 13003 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑1) = (𝐺‘𝑥)) |
31 | 30 | mpteq2dva 4744 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) |
32 | 28 | feqmptd 6249 |
. . . . . . 7
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) |
33 | 31, 32 | eqtr4d 2659 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = 𝐺) |
34 | 33 | fveq2d 6195 |
. . . . 5
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (deg‘𝐺)) |
35 | | dgrcolem1.1 |
. . . . 5
⊢ 𝑁 = (deg‘𝐺) |
36 | 34, 35 | syl6eqr 2674 |
. . . 4
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = 𝑁) |
37 | | dgrcolem1.3 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
38 | 37 | nncnd 11036 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℂ) |
39 | 38 | mulid2d 10058 |
. . . 4
⊢ (𝜑 → (1 · 𝑁) = 𝑁) |
40 | 36, 39 | eqtr4d 2659 |
. . 3
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)) |
41 | 29 | adantlr 751 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ) |
42 | | nnnn0 11299 |
. . . . . . . . . . . . . 14
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℕ0) |
43 | 42 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ0) |
44 | 43 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑑 ∈ ℕ0) |
45 | 41, 44 | expp1d 13009 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑(𝑑 + 1)) = (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))) |
46 | 45 | mpteq2dva 4744 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))) |
47 | | cnex 10017 |
. . . . . . . . . . . 12
⊢ ℂ
∈ V |
48 | 47 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈
V) |
49 | | ovexd 6680 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑑) ∈ V) |
50 | | eqidd 2623 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
51 | 32 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) |
52 | 48, 49, 41, 50, 51 | offval2 6914 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))) |
53 | 46, 52 | eqtr4d 2659 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺)) |
54 | 53 | fveq2d 6195 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺))) |
55 | 54 | adantr 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺))) |
56 | | nncn 11028 |
. . . . . . . . . . . 12
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℂ) |
57 | 56 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℂ) |
58 | | 1cnd 10056 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 1 ∈
ℂ) |
59 | 38 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℂ) |
60 | 57, 58, 59 | adddird 10065 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + (1 · 𝑁))) |
61 | 59 | mulid2d 10058 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (1 · 𝑁) = 𝑁) |
62 | 61 | oveq2d 6666 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 · 𝑁) + (1 · 𝑁)) = ((𝑑 · 𝑁) + 𝑁)) |
63 | 60, 62 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
64 | 63 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
65 | | eqidd 2623 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑑))) |
66 | | oveq1 6657 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑑) = ((𝐺‘𝑥)↑𝑑)) |
67 | 41, 51, 65, 66 | fmptco 6396 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
68 | | ssid 3624 |
. . . . . . . . . . . . . . 15
⊢ ℂ
⊆ ℂ |
69 | 68 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ⊆
ℂ) |
70 | | plypow 23961 |
. . . . . . . . . . . . . 14
⊢ ((ℂ
⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑑 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈
(Poly‘ℂ)) |
71 | 69, 58, 43, 70 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈
(Poly‘ℂ)) |
72 | | plyssc 23956 |
. . . . . . . . . . . . . 14
⊢
(Poly‘𝑆)
⊆ (Poly‘ℂ) |
73 | 26 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆)) |
74 | 72, 73 | sseldi 3601 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈
(Poly‘ℂ)) |
75 | | addcl 10018 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ) |
76 | 75 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ) |
77 | | mulcl 10020 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ) |
78 | 77 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ) |
79 | 71, 74, 76, 78 | plyco 23997 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) ∈
(Poly‘ℂ)) |
80 | 67, 79 | eqeltrrd 2702 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈
(Poly‘ℂ)) |
81 | 80 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈
(Poly‘ℂ)) |
82 | | simpr 477 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) |
83 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ) |
84 | 37 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℕ) |
85 | 83, 84 | nnmulcld 11068 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ∈ ℕ) |
86 | 85 | nnne0d 11065 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ≠ 0) |
87 | 86 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑑 · 𝑁) ≠ 0) |
88 | 82, 87 | eqnetrd 2861 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0) |
89 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0𝑝 →
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) =
(deg‘0𝑝)) |
90 | | dgr0 24018 |
. . . . . . . . . . . . 13
⊢
(deg‘0𝑝) = 0 |
91 | 89, 90 | syl6eq 2672 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0𝑝 →
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = 0) |
92 | 91 | necon3i 2826 |
. . . . . . . . . . 11
⊢
((deg‘(𝑥
∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠
0𝑝) |
93 | 88, 92 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠
0𝑝) |
94 | 74 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ∈
(Poly‘ℂ)) |
95 | 37 | nnne0d 11065 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ≠ 0) |
96 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺 = 0𝑝 →
(deg‘𝐺) =
(deg‘0𝑝)) |
97 | 96, 90 | syl6eq 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝐺 = 0𝑝 →
(deg‘𝐺) =
0) |
98 | 35, 97 | syl5eq 2668 |
. . . . . . . . . . . . . 14
⊢ (𝐺 = 0𝑝 →
𝑁 = 0) |
99 | 98 | necon3i 2826 |
. . . . . . . . . . . . 13
⊢ (𝑁 ≠ 0 → 𝐺 ≠
0𝑝) |
100 | 95, 99 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ≠
0𝑝) |
101 | 100 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ≠
0𝑝) |
102 | 101 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ≠
0𝑝) |
103 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
104 | 103, 35 | dgrmul 24026 |
. . . . . . . . . 10
⊢ ((((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈ (Poly‘ℂ) ∧ (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠ 0𝑝) ∧ (𝐺 ∈ (Poly‘ℂ)
∧ 𝐺 ≠
0𝑝)) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺)) = ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁)) |
105 | 81, 93, 94, 102, 104 | syl22anc 1327 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺)) = ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁)) |
106 | | oveq1 6657 |
. . . . . . . . . 10
⊢
((deg‘(𝑥
∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
107 | 106 | adantl 482 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
108 | 105, 107 | eqtrd 2656 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺)) = ((𝑑 · 𝑁) + 𝑁)) |
109 | 64, 108 | eqtr4d 2659 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺))) |
110 | 55, 109 | eqtr4d 2659 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)) |
111 | 110 | ex 450 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))) |
112 | 111 | expcom 451 |
. . . 4
⊢ (𝑑 ∈ ℕ → (𝜑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))) |
113 | 112 | a2d 29 |
. . 3
⊢ (𝑑 ∈ ℕ → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))) |
114 | 7, 13, 19, 25, 40, 113 | nnind 11038 |
. 2
⊢ (𝑀 ∈ ℕ → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))) |
115 | 1, 114 | mpcom 38 |
1
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)) |