Step | Hyp | Ref
| Expression |
1 | | dgrcolem1.2 |
. 2
⊢ (𝜑 → 𝑀 ∈ ℕ) |
2 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑦 = 1 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑1)) |
3 | 2 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑦 = 1 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) |
4 | 3 | fveq2d 6107 |
. . . . 5
⊢ (𝑦 = 1 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)))) |
5 | | oveq1 6556 |
. . . . 5
⊢ (𝑦 = 1 → (𝑦 · 𝑁) = (1 · 𝑁)) |
6 | 4, 5 | eqeq12d 2625 |
. . . 4
⊢ (𝑦 = 1 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))) |
7 | 6 | imbi2d 329 |
. . 3
⊢ (𝑦 = 1 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)))) |
8 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑦 = 𝑑 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑑)) |
9 | 8 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑦 = 𝑑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
10 | 9 | fveq2d 6107 |
. . . . 5
⊢ (𝑦 = 𝑑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))) |
11 | | oveq1 6556 |
. . . . 5
⊢ (𝑦 = 𝑑 → (𝑦 · 𝑁) = (𝑑 · 𝑁)) |
12 | 10, 11 | eqeq12d 2625 |
. . . 4
⊢ (𝑦 = 𝑑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))) |
13 | 12 | imbi2d 329 |
. . 3
⊢ (𝑦 = 𝑑 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)))) |
14 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑦 = (𝑑 + 1) → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑(𝑑 + 1))) |
15 | 14 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑦 = (𝑑 + 1) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) |
16 | 15 | fveq2d 6107 |
. . . . 5
⊢ (𝑦 = (𝑑 + 1) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))))) |
17 | | oveq1 6556 |
. . . . 5
⊢ (𝑦 = (𝑑 + 1) → (𝑦 · 𝑁) = ((𝑑 + 1) · 𝑁)) |
18 | 16, 17 | eqeq12d 2625 |
. . . 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 329 |
. . 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 6557 |
. . . . . . 7
⊢ (𝑦 = 𝑀 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑀)) |
21 | 20 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑦 = 𝑀 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) |
22 | 21 | fveq2d 6107 |
. . . . 5
⊢ (𝑦 = 𝑀 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) |
23 | | oveq1 6556 |
. . . . 5
⊢ (𝑦 = 𝑀 → (𝑦 · 𝑁) = (𝑀 · 𝑁)) |
24 | 22, 23 | eqeq12d 2625 |
. . . 4
⊢ (𝑦 = 𝑀 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))) |
25 | 24 | imbi2d 329 |
. . 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 23758 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) |
28 | 26, 27 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺:ℂ⟶ℂ) |
29 | 28 | ffvelrnda 6267 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ) |
30 | 29 | exp1d 12865 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑1) = (𝐺‘𝑥)) |
31 | 30 | mpteq2dva 4672 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) |
32 | 28 | feqmptd 6159 |
. . . . . . 7
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) |
33 | 31, 32 | eqtr4d 2647 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = 𝐺) |
34 | 33 | fveq2d 6107 |
. . . . 5
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (deg‘𝐺)) |
35 | | dgrcolem1.1 |
. . . . 5
⊢ 𝑁 = (deg‘𝐺) |
36 | 34, 35 | syl6eqr 2662 |
. . . 4
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = 𝑁) |
37 | | dgrcolem1.3 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
38 | 37 | nncnd 10913 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℂ) |
39 | 38 | mulid2d 9937 |
. . . 4
⊢ (𝜑 → (1 · 𝑁) = 𝑁) |
40 | 36, 39 | eqtr4d 2647 |
. . 3
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)) |
41 | 29 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ) |
42 | | nnnn0 11176 |
. . . . . . . . . . . . . 14
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℕ0) |
43 | 42 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ0) |
44 | 43 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → 𝑑 ∈ ℕ0) |
45 | 41, 44 | expp1d 12871 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑(𝑑 + 1)) = (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))) |
46 | 45 | mpteq2dva 4672 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))) |
47 | | cnex 9896 |
. . . . . . . . . . . 12
⊢ ℂ
∈ V |
48 | 47 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈
V) |
49 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑥)↑𝑑) ∈ V |
50 | 49 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑑) ∈ V) |
51 | | eqidd 2611 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
52 | 32 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) |
53 | 48, 50, 41, 51, 52 | offval2 6812 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))) |
54 | 46, 53 | eqtr4d 2647 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺)) |
55 | 54 | fveq2d 6107 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺))) |
56 | 55 | adantr 480 |
. . . . . . 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))) |
57 | | nncn 10905 |
. . . . . . . . . . . 12
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℂ) |
58 | 57 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℂ) |
59 | | 1cnd 9935 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 1 ∈
ℂ) |
60 | 38 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℂ) |
61 | 58, 59, 60 | adddird 9944 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + (1 · 𝑁))) |
62 | 60 | mulid2d 9937 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (1 · 𝑁) = 𝑁) |
63 | 62 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 · 𝑁) + (1 · 𝑁)) = ((𝑑 · 𝑁) + 𝑁)) |
64 | 61, 63 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
65 | 64 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
66 | | eqidd 2611 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑑))) |
67 | | oveq1 6556 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑑) = ((𝐺‘𝑥)↑𝑑)) |
68 | 41, 52, 66, 67 | fmptco 6303 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
69 | | ssid 3587 |
. . . . . . . . . . . . . . 15
⊢ ℂ
⊆ ℂ |
70 | 69 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ⊆
ℂ) |
71 | | plypow 23765 |
. . . . . . . . . . . . . 14
⊢ ((ℂ
⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑑 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈
(Poly‘ℂ)) |
72 | 70, 59, 43, 71 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈
(Poly‘ℂ)) |
73 | | plyssc 23760 |
. . . . . . . . . . . . . 14
⊢
(Poly‘𝑆)
⊆ (Poly‘ℂ) |
74 | 26 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆)) |
75 | 73, 74 | sseldi 3566 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈
(Poly‘ℂ)) |
76 | | addcl 9897 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ) |
77 | 76 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ) |
78 | | mulcl 9899 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ) |
79 | 78 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ) |
80 | 72, 75, 77, 79 | plyco 23801 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) ∈
(Poly‘ℂ)) |
81 | 68, 80 | eqeltrrd 2689 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈
(Poly‘ℂ)) |
82 | 81 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈
(Poly‘ℂ)) |
83 | | simpr 476 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) |
84 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ) |
85 | 37 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℕ) |
86 | 84, 85 | nnmulcld 10945 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ∈ ℕ) |
87 | 86 | nnne0d 10942 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ≠ 0) |
88 | 87 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑑 · 𝑁) ≠ 0) |
89 | 83, 88 | eqnetrd 2849 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0) |
90 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0𝑝 →
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) =
(deg‘0𝑝)) |
91 | | dgr0 23822 |
. . . . . . . . . . . . 13
⊢
(deg‘0𝑝) = 0 |
92 | 90, 91 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0𝑝 →
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = 0) |
93 | 92 | necon3i 2814 |
. . . . . . . . . . 11
⊢
((deg‘(𝑥
∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠
0𝑝) |
94 | 89, 93 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠
0𝑝) |
95 | 75 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ∈
(Poly‘ℂ)) |
96 | 37 | nnne0d 10942 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ≠ 0) |
97 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺 = 0𝑝 →
(deg‘𝐺) =
(deg‘0𝑝)) |
98 | 97, 91 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ (𝐺 = 0𝑝 →
(deg‘𝐺) =
0) |
99 | 35, 98 | syl5eq 2656 |
. . . . . . . . . . . . . 14
⊢ (𝐺 = 0𝑝 →
𝑁 = 0) |
100 | 99 | necon3i 2814 |
. . . . . . . . . . . . 13
⊢ (𝑁 ≠ 0 → 𝐺 ≠
0𝑝) |
101 | 96, 100 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ≠
0𝑝) |
102 | 101 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ≠
0𝑝) |
103 | 102 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ≠
0𝑝) |
104 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) |
105 | 104, 35 | dgrmul 23830 |
. . . . . . . . . 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)) |
106 | 82, 94, 95, 103, 105 | syl22anc 1319 |
. . . . . . . . 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)) |
107 | | oveq1 6556 |
. . . . . . . . . 10
⊢
((deg‘(𝑥
∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
108 | 107 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁)) |
109 | 106, 108 | eqtrd 2644 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺)) = ((𝑑 · 𝑁) + 𝑁)) |
110 | 65, 109 | eqtr4d 2647 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘𝑓 · 𝐺))) |
111 | 56, 110 | eqtr4d 2647 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)) |
112 | 111 | ex 449 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))) |
113 | 112 | expcom 450 |
. . . 4
⊢ (𝑑 ∈ ℕ → (𝜑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))) |
114 | 113 | 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)))) |
115 | 7, 13, 19, 25, 40, 114 | nnind 10915 |
. 2
⊢ (𝑀 ∈ ℕ → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))) |
116 | 1, 115 | mpcom 37 |
1
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)) |