| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | dgrcl 26273 | . . . . . . 7
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈
ℕ0) | 
| 2 |  | dgrcl 26273 | . . . . . . 7
⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈
ℕ0) | 
| 3 |  | nn0addcl 12563 | . . . . . . 7
⊢
(((deg‘𝐹)
∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) →
((deg‘𝐹) +
(deg‘𝐺)) ∈
ℕ0) | 
| 4 | 1, 2, 3 | syl2an 596 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈
ℕ0) | 
| 5 |  | c0ex 11256 | . . . . . . 7
⊢ 0 ∈
V | 
| 6 | 5 | fvconst2 7225 | . . . . . 6
⊢
(((deg‘𝐹) +
(deg‘𝐺)) ∈
ℕ0 → ((ℕ0 ×
{0})‘((deg‘𝐹) +
(deg‘𝐺))) =
0) | 
| 7 | 4, 6 | syl 17 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℕ0 ×
{0})‘((deg‘𝐹) +
(deg‘𝐺))) =
0) | 
| 8 |  | fveq2 6905 | . . . . . . . 8
⊢ ((𝐹 ∘f ·
𝐺) = 0𝑝
→ (coeff‘(𝐹
∘f · 𝐺)) =
(coeff‘0𝑝)) | 
| 9 |  | coe0 26296 | . . . . . . . 8
⊢
(coeff‘0𝑝) = (ℕ0 ×
{0}) | 
| 10 | 8, 9 | eqtrdi 2792 | . . . . . . 7
⊢ ((𝐹 ∘f ·
𝐺) = 0𝑝
→ (coeff‘(𝐹
∘f · 𝐺)) = (ℕ0 ×
{0})) | 
| 11 | 10 | fveq1d 6907 | . . . . . 6
⊢ ((𝐹 ∘f ·
𝐺) = 0𝑝
→ ((coeff‘(𝐹
∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 ×
{0})‘((deg‘𝐹) +
(deg‘𝐺)))) | 
| 12 | 11 | eqeq1d 2738 | . . . . 5
⊢ ((𝐹 ∘f ·
𝐺) = 0𝑝
→ (((coeff‘(𝐹
∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ ((ℕ0 ×
{0})‘((deg‘𝐹) +
(deg‘𝐺))) =
0)) | 
| 13 | 7, 12 | syl5ibrcom 247 | . . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 →
((coeff‘(𝐹
∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0)) | 
| 14 |  | eqid 2736 | . . . . . . 7
⊢
(coeff‘𝐹) =
(coeff‘𝐹) | 
| 15 |  | eqid 2736 | . . . . . . 7
⊢
(coeff‘𝐺) =
(coeff‘𝐺) | 
| 16 |  | eqid 2736 | . . . . . . 7
⊢
(deg‘𝐹) =
(deg‘𝐹) | 
| 17 |  | eqid 2736 | . . . . . . 7
⊢
(deg‘𝐺) =
(deg‘𝐺) | 
| 18 | 14, 15, 16, 17 | coemulhi 26294 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺)))) | 
| 19 | 18 | eqeq1d 2738 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))) = 0)) | 
| 20 | 14 | coef3 26272 | . . . . . . . 8
⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ) | 
| 21 | 20 | adantr 480 | . . . . . . 7
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ) | 
| 22 | 1 | adantr 480 | . . . . . . 7
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈
ℕ0) | 
| 23 | 21, 22 | ffvelcdmd 7104 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ) | 
| 24 | 15 | coef3 26272 | . . . . . . . 8
⊢ (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ) | 
| 25 | 24 | adantl 481 | . . . . . . 7
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ) | 
| 26 | 2 | adantl 481 | . . . . . . 7
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈
ℕ0) | 
| 27 | 25, 26 | ffvelcdmd 7104 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ) | 
| 28 | 23, 27 | mul0ord 11914 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))) | 
| 29 | 19, 28 | bitrd 279 | . . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))) | 
| 30 | 13, 29 | sylibd 239 | . . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 →
(((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))) | 
| 31 | 16, 14 | dgreq0 26306 | . . . . 5
⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔
((coeff‘𝐹)‘(deg‘𝐹)) = 0)) | 
| 32 | 31 | adantr 480 | . . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔
((coeff‘𝐹)‘(deg‘𝐹)) = 0)) | 
| 33 | 17, 15 | dgreq0 26306 | . . . . 5
⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔
((coeff‘𝐺)‘(deg‘𝐺)) = 0)) | 
| 34 | 33 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔
((coeff‘𝐺)‘(deg‘𝐺)) = 0)) | 
| 35 | 32, 34 | orbi12d 918 | . . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) ↔
(((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))) | 
| 36 | 30, 35 | sylibrd 259 | . 2
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 →
(𝐹 = 0𝑝
∨ 𝐺 =
0𝑝))) | 
| 37 |  | cnex 11237 | . . . . . . 7
⊢ ℂ
∈ V | 
| 38 | 37 | a1i 11 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V) | 
| 39 |  | plyf 26238 | . . . . . . 7
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | 
| 40 | 39 | adantl 481 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ) | 
| 41 |  | 0cnd 11255 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ) | 
| 42 |  | mul02 11440 | . . . . . . 7
⊢ (𝑥 ∈ ℂ → (0
· 𝑥) =
0) | 
| 43 | 42 | adantl 481 | . . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) | 
| 44 | 38, 40, 41, 41, 43 | caofid2 7734 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0})
∘f · 𝐺) = (ℂ × {0})) | 
| 45 |  | id 22 | . . . . . . . 8
⊢ (𝐹 = 0𝑝 →
𝐹 =
0𝑝) | 
| 46 |  | df-0p 25706 | . . . . . . . 8
⊢
0𝑝 = (ℂ × {0}) | 
| 47 | 45, 46 | eqtrdi 2792 | . . . . . . 7
⊢ (𝐹 = 0𝑝 →
𝐹 = (ℂ ×
{0})) | 
| 48 | 47 | oveq1d 7447 | . . . . . 6
⊢ (𝐹 = 0𝑝 →
(𝐹 ∘f
· 𝐺) = ((ℂ
× {0}) ∘f · 𝐺)) | 
| 49 | 48 | eqeq1d 2738 | . . . . 5
⊢ (𝐹 = 0𝑝 →
((𝐹 ∘f
· 𝐺) = (ℂ
× {0}) ↔ ((ℂ × {0}) ∘f · 𝐺) = (ℂ ×
{0}))) | 
| 50 | 44, 49 | syl5ibrcom 247 | . . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹 ∘f ·
𝐺) = (ℂ ×
{0}))) | 
| 51 |  | plyf 26238 | . . . . . . 7
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | 
| 52 | 51 | adantr 480 | . . . . . 6
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ) | 
| 53 |  | mul01 11441 | . . . . . . 7
⊢ (𝑥 ∈ ℂ → (𝑥 · 0) =
0) | 
| 54 | 53 | adantl 481 | . . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0) | 
| 55 | 38, 52, 41, 41, 54 | caofid1 7733 | . . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · (ℂ
× {0})) = (ℂ × {0})) | 
| 56 |  | id 22 | . . . . . . . 8
⊢ (𝐺 = 0𝑝 →
𝐺 =
0𝑝) | 
| 57 | 56, 46 | eqtrdi 2792 | . . . . . . 7
⊢ (𝐺 = 0𝑝 →
𝐺 = (ℂ ×
{0})) | 
| 58 | 57 | oveq2d 7448 | . . . . . 6
⊢ (𝐺 = 0𝑝 →
(𝐹 ∘f
· 𝐺) = (𝐹 ∘f ·
(ℂ × {0}))) | 
| 59 | 58 | eqeq1d 2738 | . . . . 5
⊢ (𝐺 = 0𝑝 →
((𝐹 ∘f
· 𝐺) = (ℂ
× {0}) ↔ (𝐹
∘f · (ℂ × {0})) = (ℂ ×
{0}))) | 
| 60 | 55, 59 | syl5ibrcom 247 | . . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹 ∘f ·
𝐺) = (ℂ ×
{0}))) | 
| 61 | 50, 60 | jaod 859 | . . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) →
(𝐹 ∘f
· 𝐺) = (ℂ
× {0}))) | 
| 62 | 46 | eqeq2i 2749 | . . 3
⊢ ((𝐹 ∘f ·
𝐺) = 0𝑝
↔ (𝐹
∘f · 𝐺) = (ℂ × {0})) | 
| 63 | 61, 62 | imbitrrdi 252 | . 2
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) →
(𝐹 ∘f
· 𝐺) =
0𝑝)) | 
| 64 | 36, 63 | impbid 212 | 1
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 ↔
(𝐹 = 0𝑝
∨ 𝐺 =
0𝑝))) |