| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | dgrcolem1.2 | . 2
⊢ (𝜑 → 𝑀 ∈ ℕ) | 
| 2 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑦 = 1 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑1)) | 
| 3 | 2 | mpteq2dv 5244 | . . . . . 6
⊢ (𝑦 = 1 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) | 
| 4 | 3 | fveq2d 6910 | . . . . 5
⊢ (𝑦 = 1 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)))) | 
| 5 |  | oveq1 7438 | . . . . 5
⊢ (𝑦 = 1 → (𝑦 · 𝑁) = (1 · 𝑁)) | 
| 6 | 4, 5 | eqeq12d 2753 | . . . 4
⊢ (𝑦 = 1 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁))) | 
| 7 | 6 | imbi2d 340 | . . 3
⊢ (𝑦 = 1 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)))) | 
| 8 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑦 = 𝑑 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑑)) | 
| 9 | 8 | mpteq2dv 5244 | . . . . . 6
⊢ (𝑦 = 𝑑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) | 
| 10 | 9 | fveq2d 6910 | . . . . 5
⊢ (𝑦 = 𝑑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)))) | 
| 11 |  | oveq1 7438 | . . . . 5
⊢ (𝑦 = 𝑑 → (𝑦 · 𝑁) = (𝑑 · 𝑁)) | 
| 12 | 10, 11 | eqeq12d 2753 | . . . 4
⊢ (𝑦 = 𝑑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁))) | 
| 13 | 12 | imbi2d 340 | . . 3
⊢ (𝑦 = 𝑑 → ((𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁)) ↔ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)))) | 
| 14 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑦 = (𝑑 + 1) → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑(𝑑 + 1))) | 
| 15 | 14 | mpteq2dv 5244 | . . . . . 6
⊢ (𝑦 = (𝑑 + 1) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) | 
| 16 | 15 | fveq2d 6910 | . . . . 5
⊢ (𝑦 = (𝑑 + 1) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))))) | 
| 17 |  | oveq1 7438 | . . . . 5
⊢ (𝑦 = (𝑑 + 1) → (𝑦 · 𝑁) = ((𝑑 + 1) · 𝑁)) | 
| 18 | 16, 17 | eqeq12d 2753 | . . . 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 340 | . . 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 7439 | . . . . . . 7
⊢ (𝑦 = 𝑀 → ((𝐺‘𝑥)↑𝑦) = ((𝐺‘𝑥)↑𝑀)) | 
| 21 | 20 | mpteq2dv 5244 | . . . . . 6
⊢ (𝑦 = 𝑀 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) | 
| 22 | 21 | fveq2d 6910 | . . . . 5
⊢ (𝑦 = 𝑀 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀)))) | 
| 23 |  | oveq1 7438 | . . . . 5
⊢ (𝑦 = 𝑀 → (𝑦 · 𝑁) = (𝑀 · 𝑁)) | 
| 24 | 22, 23 | eqeq12d 2753 | . . . 4
⊢ (𝑦 = 𝑀 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑦))) = (𝑦 · 𝑁) ↔ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))) | 
| 25 | 24 | imbi2d 340 | . . 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 26237 | . . . . . . . . . . 11
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | 
| 28 | 26, 27 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝐺:ℂ⟶ℂ) | 
| 29 | 28 | ffvelcdmda 7104 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ) | 
| 30 | 29 | exp1d 14181 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑1) = (𝐺‘𝑥)) | 
| 31 | 30 | mpteq2dva 5242 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) | 
| 32 | 28 | feqmptd 6977 | . . . . . . 7
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) | 
| 33 | 31, 32 | eqtr4d 2780 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1)) = 𝐺) | 
| 34 | 33 | fveq2d 6910 | . . . . 5
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (deg‘𝐺)) | 
| 35 |  | dgrcolem1.1 | . . . . 5
⊢ 𝑁 = (deg‘𝐺) | 
| 36 | 34, 35 | eqtr4di 2795 | . . . 4
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = 𝑁) | 
| 37 |  | dgrcolem1.3 | . . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 38 | 37 | nncnd 12282 | . . . . 5
⊢ (𝜑 → 𝑁 ∈ ℂ) | 
| 39 | 38 | mullidd 11279 | . . . 4
⊢ (𝜑 → (1 · 𝑁) = 𝑁) | 
| 40 | 36, 39 | eqtr4d 2780 | . . 3
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑1))) = (1 · 𝑁)) | 
| 41 | 29 | adantlr 715 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → (𝐺‘𝑥) ∈ ℂ) | 
| 42 |  | nnnn0 12533 | . . . . . . . . . . . . . 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 14187 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑(𝑑 + 1)) = (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥))) | 
| 46 | 45 | mpteq2dva 5242 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))) | 
| 47 |  | cnex 11236 | . . . . . . . . . . . 12
⊢ ℂ
∈ V | 
| 48 | 47 | a1i 11 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈
V) | 
| 49 |  | ovexd 7466 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑥 ∈ ℂ) → ((𝐺‘𝑥)↑𝑑) ∈ V) | 
| 50 |  | eqidd 2738 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) | 
| 51 | 32 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑥 ∈ ℂ ↦ (𝐺‘𝑥))) | 
| 52 | 48, 49, 41, 50, 51 | offval2 7717 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘f · 𝐺) = (𝑥 ∈ ℂ ↦ (((𝐺‘𝑥)↑𝑑) · (𝐺‘𝑥)))) | 
| 53 | 46, 52 | eqtr4d 2780 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1))) = ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘f · 𝐺)) | 
| 54 | 53 | fveq2d 6910 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘f · 𝐺))) | 
| 55 | 54 | 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))) | 
| 56 |  | oveq1 7438 | . . . . . . . . 9
⊢
((deg‘(𝑥
∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁)) | 
| 57 | 56 | adantl 481 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) + 𝑁) = ((𝑑 · 𝑁) + 𝑁)) | 
| 58 |  | eqidd 2738 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) = (𝑦 ∈ ℂ ↦ (𝑦↑𝑑))) | 
| 59 |  | oveq1 7438 | . . . . . . . . . . . 12
⊢ (𝑦 = (𝐺‘𝑥) → (𝑦↑𝑑) = ((𝐺‘𝑥)↑𝑑)) | 
| 60 | 41, 51, 58, 59 | fmptco 7149 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) = (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) | 
| 61 |  | ssidd 4007 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ⊆
ℂ) | 
| 62 |  | 1cnd 11256 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 1 ∈
ℂ) | 
| 63 |  | plypow 26244 | . . . . . . . . . . . . 13
⊢ ((ℂ
⊆ ℂ ∧ 1 ∈ ℂ ∧ 𝑑 ∈ ℕ0) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈
(Poly‘ℂ)) | 
| 64 | 61, 62, 43, 63 | syl3anc 1373 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∈
(Poly‘ℂ)) | 
| 65 |  | plyssc 26239 | . . . . . . . . . . . . 13
⊢
(Poly‘𝑆)
⊆ (Poly‘ℂ) | 
| 66 | 26 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈ (Poly‘𝑆)) | 
| 67 | 65, 66 | sselid 3981 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ∈
(Poly‘ℂ)) | 
| 68 |  | addcl 11237 | . . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 + 𝑤) ∈ ℂ) | 
| 69 | 68 | adantl 481 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 + 𝑤) ∈ ℂ) | 
| 70 |  | mulcl 11239 | . . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ) | 
| 71 | 70 | adantl 481 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ) | 
| 72 | 64, 67, 69, 71 | plyco 26280 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑦 ∈ ℂ ↦ (𝑦↑𝑑)) ∘ 𝐺) ∈
(Poly‘ℂ)) | 
| 73 | 60, 72 | eqeltrrd 2842 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈
(Poly‘ℂ)) | 
| 74 | 73 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∈
(Poly‘ℂ)) | 
| 75 |  | simpr 484 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) | 
| 76 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℕ) | 
| 77 | 37 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℕ) | 
| 78 | 76, 77 | nnmulcld 12319 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ∈ ℕ) | 
| 79 | 78 | nnne0d 12316 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑑 · 𝑁) ≠ 0) | 
| 80 | 79 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑑 · 𝑁) ≠ 0) | 
| 81 | 75, 80 | eqnetrd 3008 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0) | 
| 82 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0𝑝 →
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) =
(deg‘0𝑝)) | 
| 83 |  | dgr0 26302 | . . . . . . . . . . . 12
⊢
(deg‘0𝑝) = 0 | 
| 84 | 82, 83 | eqtrdi 2793 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) = 0𝑝 →
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = 0) | 
| 85 | 84 | necon3i 2973 | . . . . . . . . . 10
⊢
((deg‘(𝑥
∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) ≠ 0 → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠
0𝑝) | 
| 86 | 81, 85 | syl 17 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ≠
0𝑝) | 
| 87 | 67 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ∈
(Poly‘ℂ)) | 
| 88 | 37 | nnne0d 12316 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ≠ 0) | 
| 89 |  | fveq2 6906 | . . . . . . . . . . . . . . 15
⊢ (𝐺 = 0𝑝 →
(deg‘𝐺) =
(deg‘0𝑝)) | 
| 90 | 89, 83 | eqtrdi 2793 | . . . . . . . . . . . . . 14
⊢ (𝐺 = 0𝑝 →
(deg‘𝐺) =
0) | 
| 91 | 35, 90 | eqtrid 2789 | . . . . . . . . . . . . 13
⊢ (𝐺 = 0𝑝 →
𝑁 = 0) | 
| 92 | 91 | necon3i 2973 | . . . . . . . . . . . 12
⊢ (𝑁 ≠ 0 → 𝐺 ≠
0𝑝) | 
| 93 | 88, 92 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ≠
0𝑝) | 
| 94 | 93 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 ≠
0𝑝) | 
| 95 | 94 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → 𝐺 ≠
0𝑝) | 
| 96 |  | eqid 2737 | . . . . . . . . . 10
⊢
(deg‘(𝑥 ∈
ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) | 
| 97 | 96, 35 | dgrmul 26310 | . . . . . . . . 9
⊢ ((((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)) | 
| 98 | 74, 86, 87, 95, 97 | syl22anc 839 | . . . . . . . 8
⊢ (((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)) | 
| 99 |  | nncn 12274 | . . . . . . . . . . 11
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℂ) | 
| 100 | 99 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑑 ∈ ℂ) | 
| 101 | 38 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝑁 ∈ ℂ) | 
| 102 | 100, 101 | adddirp1d 11287 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁)) | 
| 103 | 102 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = ((𝑑 · 𝑁) + 𝑁)) | 
| 104 | 57, 98, 103 | 3eqtr4rd 2788 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → ((𝑑 + 1) · 𝑁) = (deg‘((𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑)) ∘f · 𝐺))) | 
| 105 | 55, 104 | eqtr4d 2780 | . . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁)) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)) | 
| 106 | 105 | ex 412 | . . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁))) | 
| 107 | 106 | expcom 413 | . . . 4
⊢ (𝑑 ∈ ℕ → (𝜑 → ((deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑑))) = (𝑑 · 𝑁) → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑(𝑑 + 1)))) = ((𝑑 + 1) · 𝑁)))) | 
| 108 | 107 | 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)))) | 
| 109 | 7, 13, 19, 25, 40, 108 | nnind 12284 | . 2
⊢ (𝑀 ∈ ℕ → (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))) | 
| 110 | 1, 109 | mpcom 38 | 1
⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)) |