| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | plydiv.f | . . 3
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | 
| 2 |  | plybss 26233 | . . 3
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | 
| 3 |  | ply0 26247 | . . 3
⊢ (𝑆 ⊆ ℂ →
0𝑝 ∈ (Poly‘𝑆)) | 
| 4 | 1, 2, 3 | 3syl 18 | . 2
⊢ (𝜑 → 0𝑝
∈ (Poly‘𝑆)) | 
| 5 |  | plydiv.0 | . . 3
⊢ (𝜑 → (𝐹 = 0𝑝 ∨
((deg‘𝐹) −
(deg‘𝐺)) <
0)) | 
| 6 |  | cnex 11236 | . . . . . . 7
⊢ ℂ
∈ V | 
| 7 | 6 | a1i 11 | . . . . . 6
⊢ (𝜑 → ℂ ∈
V) | 
| 8 |  | plyf 26237 | . . . . . . 7
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | 
| 9 |  | ffn 6736 | . . . . . . 7
⊢ (𝐹:ℂ⟶ℂ →
𝐹 Fn
ℂ) | 
| 10 | 1, 8, 9 | 3syl 18 | . . . . . 6
⊢ (𝜑 → 𝐹 Fn ℂ) | 
| 11 |  | plydiv.g | . . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | 
| 12 |  | plyf 26237 | . . . . . . . 8
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | 
| 13 |  | ffn 6736 | . . . . . . . 8
⊢ (𝐺:ℂ⟶ℂ →
𝐺 Fn
ℂ) | 
| 14 | 11, 12, 13 | 3syl 18 | . . . . . . 7
⊢ (𝜑 → 𝐺 Fn ℂ) | 
| 15 |  | plyf 26237 | . . . . . . . 8
⊢
(0𝑝 ∈ (Poly‘𝑆) →
0𝑝:ℂ⟶ℂ) | 
| 16 |  | ffn 6736 | . . . . . . . 8
⊢
(0𝑝:ℂ⟶ℂ →
0𝑝 Fn ℂ) | 
| 17 | 4, 15, 16 | 3syl 18 | . . . . . . 7
⊢ (𝜑 → 0𝑝 Fn
ℂ) | 
| 18 |  | inidm 4227 | . . . . . . 7
⊢ (ℂ
∩ ℂ) = ℂ | 
| 19 | 14, 17, 7, 7, 18 | offn 7710 | . . . . . 6
⊢ (𝜑 → (𝐺 ∘f ·
0𝑝) Fn ℂ) | 
| 20 |  | eqidd 2738 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐹‘𝑧) = (𝐹‘𝑧)) | 
| 21 |  | eqidd 2738 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) = (𝐺‘𝑧)) | 
| 22 |  | 0pval 25706 | . . . . . . . . 9
⊢ (𝑧 ∈ ℂ →
(0𝑝‘𝑧) = 0) | 
| 23 | 22 | adantl 481 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) →
(0𝑝‘𝑧) = 0) | 
| 24 | 14, 17, 7, 7, 18, 21, 23 | ofval 7708 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺 ∘f ·
0𝑝)‘𝑧) = ((𝐺‘𝑧) · 0)) | 
| 25 | 11, 12 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐺:ℂ⟶ℂ) | 
| 26 | 25 | ffvelcdmda 7104 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ) | 
| 27 | 26 | mul01d 11460 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧) · 0) = 0) | 
| 28 | 24, 27 | eqtrd 2777 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺 ∘f ·
0𝑝)‘𝑧) = 0) | 
| 29 | 1, 8 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝐹:ℂ⟶ℂ) | 
| 30 | 29 | ffvelcdmda 7104 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐹‘𝑧) ∈ ℂ) | 
| 31 | 30 | subid1d 11609 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐹‘𝑧) − 0) = (𝐹‘𝑧)) | 
| 32 | 7, 10, 19, 10, 20, 28, 31 | offveq 7723 | . . . . 5
⊢ (𝜑 → (𝐹 ∘f − (𝐺 ∘f ·
0𝑝)) = 𝐹) | 
| 33 | 32 | eqeq1d 2739 | . . . 4
⊢ (𝜑 → ((𝐹 ∘f − (𝐺 ∘f ·
0𝑝)) = 0𝑝 ↔ 𝐹 = 0𝑝)) | 
| 34 | 32 | fveq2d 6910 | . . . . . 6
⊢ (𝜑 → (deg‘(𝐹 ∘f −
(𝐺 ∘f
· 0𝑝))) = (deg‘𝐹)) | 
| 35 |  | dgrcl 26272 | . . . . . . . . . . 11
⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈
ℕ0) | 
| 36 | 11, 35 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → (deg‘𝐺) ∈
ℕ0) | 
| 37 | 36 | nn0red 12588 | . . . . . . . . 9
⊢ (𝜑 → (deg‘𝐺) ∈
ℝ) | 
| 38 | 37 | recnd 11289 | . . . . . . . 8
⊢ (𝜑 → (deg‘𝐺) ∈
ℂ) | 
| 39 | 38 | addlidd 11462 | . . . . . . 7
⊢ (𝜑 → (0 + (deg‘𝐺)) = (deg‘𝐺)) | 
| 40 | 39 | eqcomd 2743 | . . . . . 6
⊢ (𝜑 → (deg‘𝐺) = (0 + (deg‘𝐺))) | 
| 41 | 34, 40 | breq12d 5156 | . . . . 5
⊢ (𝜑 → ((deg‘(𝐹 ∘f −
(𝐺 ∘f
· 0𝑝))) < (deg‘𝐺) ↔ (deg‘𝐹) < (0 + (deg‘𝐺)))) | 
| 42 |  | dgrcl 26272 | . . . . . . . 8
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈
ℕ0) | 
| 43 | 1, 42 | syl 17 | . . . . . . 7
⊢ (𝜑 → (deg‘𝐹) ∈
ℕ0) | 
| 44 | 43 | nn0red 12588 | . . . . . 6
⊢ (𝜑 → (deg‘𝐹) ∈
ℝ) | 
| 45 |  | 0red 11264 | . . . . . 6
⊢ (𝜑 → 0 ∈
ℝ) | 
| 46 | 44, 37, 45 | ltsubaddd 11859 | . . . . 5
⊢ (𝜑 → (((deg‘𝐹) − (deg‘𝐺)) < 0 ↔
(deg‘𝐹) < (0 +
(deg‘𝐺)))) | 
| 47 | 41, 46 | bitr4d 282 | . . . 4
⊢ (𝜑 → ((deg‘(𝐹 ∘f −
(𝐺 ∘f
· 0𝑝))) < (deg‘𝐺) ↔ ((deg‘𝐹) − (deg‘𝐺)) < 0)) | 
| 48 | 33, 47 | orbi12d 919 | . . 3
⊢ (𝜑 → (((𝐹 ∘f − (𝐺 ∘f ·
0𝑝)) = 0𝑝 ∨ (deg‘(𝐹 ∘f −
(𝐺 ∘f
· 0𝑝))) < (deg‘𝐺)) ↔ (𝐹 = 0𝑝 ∨
((deg‘𝐹) −
(deg‘𝐺)) <
0))) | 
| 49 | 5, 48 | mpbird 257 | . 2
⊢ (𝜑 → ((𝐹 ∘f − (𝐺 ∘f ·
0𝑝)) = 0𝑝 ∨ (deg‘(𝐹 ∘f −
(𝐺 ∘f
· 0𝑝))) < (deg‘𝐺))) | 
| 50 |  | plydiv.r | . . . . . 6
⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f ·
𝑞)) | 
| 51 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑞 = 0𝑝 →
(𝐺 ∘f
· 𝑞) = (𝐺 ∘f ·
0𝑝)) | 
| 52 | 51 | oveq2d 7447 | . . . . . 6
⊢ (𝑞 = 0𝑝 →
(𝐹 ∘f
− (𝐺
∘f · 𝑞)) = (𝐹 ∘f − (𝐺 ∘f ·
0𝑝))) | 
| 53 | 50, 52 | eqtrid 2789 | . . . . 5
⊢ (𝑞 = 0𝑝 →
𝑅 = (𝐹 ∘f − (𝐺 ∘f ·
0𝑝))) | 
| 54 | 53 | eqeq1d 2739 | . . . 4
⊢ (𝑞 = 0𝑝 →
(𝑅 = 0𝑝
↔ (𝐹
∘f − (𝐺 ∘f ·
0𝑝)) = 0𝑝)) | 
| 55 | 53 | fveq2d 6910 | . . . . 5
⊢ (𝑞 = 0𝑝 →
(deg‘𝑅) =
(deg‘(𝐹
∘f − (𝐺 ∘f ·
0𝑝)))) | 
| 56 | 55 | breq1d 5153 | . . . 4
⊢ (𝑞 = 0𝑝 →
((deg‘𝑅) <
(deg‘𝐺) ↔
(deg‘(𝐹
∘f − (𝐺 ∘f ·
0𝑝))) < (deg‘𝐺))) | 
| 57 | 54, 56 | orbi12d 919 | . . 3
⊢ (𝑞 = 0𝑝 →
((𝑅 = 0𝑝
∨ (deg‘𝑅) <
(deg‘𝐺)) ↔
((𝐹 ∘f
− (𝐺
∘f · 0𝑝)) = 0𝑝
∨ (deg‘(𝐹
∘f − (𝐺 ∘f ·
0𝑝))) < (deg‘𝐺)))) | 
| 58 | 57 | rspcev 3622 | . 2
⊢
((0𝑝 ∈ (Poly‘𝑆) ∧ ((𝐹 ∘f − (𝐺 ∘f ·
0𝑝)) = 0𝑝 ∨ (deg‘(𝐹 ∘f −
(𝐺 ∘f
· 0𝑝))) < (deg‘𝐺))) → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨
(deg‘𝑅) <
(deg‘𝐺))) | 
| 59 | 4, 49, 58 | syl2anc 584 | 1
⊢ (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨
(deg‘𝑅) <
(deg‘𝐺))) |