Proof of Theorem vieta1lem1
| Step | Hyp | Ref
| Expression |
| 1 | | vieta1lem.9 |
. . 3
⊢ 𝑄 = (𝐹 quot (Xp
∘f − (ℂ × {𝑧}))) |
| 2 | | plyssc 26239 |
. . . . 5
⊢
(Poly‘𝑆)
⊆ (Poly‘ℂ) |
| 3 | | vieta1.4 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
| 4 | 3 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐹 ∈ (Poly‘𝑆)) |
| 5 | 2, 4 | sselid 3981 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐹 ∈
(Poly‘ℂ)) |
| 6 | | vieta1.3 |
. . . . . . . . 9
⊢ 𝑅 = (◡𝐹 “ {0}) |
| 7 | | cnvimass 6100 |
. . . . . . . . 9
⊢ (◡𝐹 “ {0}) ⊆ dom 𝐹 |
| 8 | 6, 7 | eqsstri 4030 |
. . . . . . . 8
⊢ 𝑅 ⊆ dom 𝐹 |
| 9 | | plyf 26237 |
. . . . . . . . 9
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) |
| 10 | 3, 9 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
| 11 | 8, 10 | fssdm 6755 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ⊆ ℂ) |
| 12 | 11 | sselda 3983 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝑧 ∈ ℂ) |
| 13 | | eqid 2737 |
. . . . . . 7
⊢
(Xp ∘f − (ℂ ×
{𝑧})) =
(Xp ∘f − (ℂ × {𝑧})) |
| 14 | 13 | plyremlem 26346 |
. . . . . 6
⊢ (𝑧 ∈ ℂ →
((Xp ∘f − (ℂ × {𝑧})) ∈ (Poly‘ℂ)
∧ (deg‘(Xp ∘f − (ℂ
× {𝑧}))) = 1 ∧
(◡(Xp
∘f − (ℂ × {𝑧})) “ {0}) = {𝑧})) |
| 15 | 12, 14 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → ((Xp
∘f − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧
(deg‘(Xp ∘f − (ℂ ×
{𝑧}))) = 1 ∧ (◡(Xp ∘f
− (ℂ × {𝑧})) “ {0}) = {𝑧})) |
| 16 | 15 | simp1d 1143 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (Xp
∘f − (ℂ × {𝑧})) ∈
(Poly‘ℂ)) |
| 17 | 15 | simp2d 1144 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (deg‘(Xp
∘f − (ℂ × {𝑧}))) = 1) |
| 18 | | ax-1ne0 11224 |
. . . . . . 7
⊢ 1 ≠
0 |
| 19 | 18 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 1 ≠ 0) |
| 20 | 17, 19 | eqnetrd 3008 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (deg‘(Xp
∘f − (ℂ × {𝑧}))) ≠ 0) |
| 21 | | fveq2 6906 |
. . . . . . 7
⊢
((Xp ∘f − (ℂ ×
{𝑧})) =
0𝑝 → (deg‘(Xp
∘f − (ℂ × {𝑧}))) =
(deg‘0𝑝)) |
| 22 | | dgr0 26302 |
. . . . . . 7
⊢
(deg‘0𝑝) = 0 |
| 23 | 21, 22 | eqtrdi 2793 |
. . . . . 6
⊢
((Xp ∘f − (ℂ ×
{𝑧})) =
0𝑝 → (deg‘(Xp
∘f − (ℂ × {𝑧}))) = 0) |
| 24 | 23 | necon3i 2973 |
. . . . 5
⊢
((deg‘(Xp ∘f − (ℂ
× {𝑧}))) ≠ 0
→ (Xp ∘f − (ℂ ×
{𝑧})) ≠
0𝑝) |
| 25 | 20, 24 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (Xp
∘f − (ℂ × {𝑧})) ≠
0𝑝) |
| 26 | | quotcl2 26344 |
. . . 4
⊢ ((𝐹 ∈ (Poly‘ℂ)
∧ (Xp ∘f − (ℂ × {𝑧})) ∈ (Poly‘ℂ)
∧ (Xp ∘f − (ℂ × {𝑧})) ≠ 0𝑝)
→ (𝐹 quot
(Xp ∘f − (ℂ × {𝑧}))) ∈
(Poly‘ℂ)) |
| 27 | 5, 16, 25, 26 | syl3anc 1373 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝐹 quot (Xp
∘f − (ℂ × {𝑧}))) ∈
(Poly‘ℂ)) |
| 28 | 1, 27 | eqeltrid 2845 |
. 2
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝑄 ∈
(Poly‘ℂ)) |
| 29 | | 1cnd 11256 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 1 ∈ ℂ) |
| 30 | | vieta1lem.6 |
. . . . 5
⊢ (𝜑 → 𝐷 ∈ ℕ) |
| 31 | 30 | nncnd 12282 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 32 | 31 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐷 ∈ ℂ) |
| 33 | | dgrcl 26272 |
. . . . 5
⊢ (𝑄 ∈ (Poly‘ℂ)
→ (deg‘𝑄) ∈
ℕ0) |
| 34 | 28, 33 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (deg‘𝑄) ∈
ℕ0) |
| 35 | 34 | nn0cnd 12589 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (deg‘𝑄) ∈ ℂ) |
| 36 | | ax-1cn 11213 |
. . . . 5
⊢ 1 ∈
ℂ |
| 37 | | addcom 11447 |
. . . . 5
⊢ ((1
∈ ℂ ∧ 𝐷
∈ ℂ) → (1 + 𝐷) = (𝐷 + 1)) |
| 38 | 36, 32, 37 | sylancr 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (1 + 𝐷) = (𝐷 + 1)) |
| 39 | | vieta1lem.7 |
. . . . . . 7
⊢ (𝜑 → (𝐷 + 1) = 𝑁) |
| 40 | | vieta1.2 |
. . . . . . 7
⊢ 𝑁 = (deg‘𝐹) |
| 41 | 39, 40 | eqtrdi 2793 |
. . . . . 6
⊢ (𝜑 → (𝐷 + 1) = (deg‘𝐹)) |
| 42 | 41 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝐷 + 1) = (deg‘𝐹)) |
| 43 | 6 | eleq2i 2833 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑅 ↔ 𝑧 ∈ (◡𝐹 “ {0})) |
| 44 | 10 | ffnd 6737 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn ℂ) |
| 45 | | fniniseg 7080 |
. . . . . . . . . . 11
⊢ (𝐹 Fn ℂ → (𝑧 ∈ (◡𝐹 “ {0}) ↔ (𝑧 ∈ ℂ ∧ (𝐹‘𝑧) = 0))) |
| 46 | 44, 45 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ (◡𝐹 “ {0}) ↔ (𝑧 ∈ ℂ ∧ (𝐹‘𝑧) = 0))) |
| 47 | 43, 46 | bitrid 283 |
. . . . . . . . 9
⊢ (𝜑 → (𝑧 ∈ 𝑅 ↔ (𝑧 ∈ ℂ ∧ (𝐹‘𝑧) = 0))) |
| 48 | 47 | simplbda 499 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝐹‘𝑧) = 0) |
| 49 | 13 | facth 26348 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑧 ∈ ℂ ∧ (𝐹‘𝑧) = 0) → 𝐹 = ((Xp
∘f − (ℂ × {𝑧})) ∘f · (𝐹 quot (Xp
∘f − (ℂ × {𝑧}))))) |
| 50 | 4, 12, 48, 49 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐹 = ((Xp
∘f − (ℂ × {𝑧})) ∘f · (𝐹 quot (Xp
∘f − (ℂ × {𝑧}))))) |
| 51 | 1 | oveq2i 7442 |
. . . . . . 7
⊢
((Xp ∘f − (ℂ ×
{𝑧})) ∘f
· 𝑄) =
((Xp ∘f − (ℂ × {𝑧})) ∘f ·
(𝐹 quot
(Xp ∘f − (ℂ × {𝑧})))) |
| 52 | 50, 51 | eqtr4di 2795 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐹 = ((Xp
∘f − (ℂ × {𝑧})) ∘f · 𝑄)) |
| 53 | 52 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (deg‘𝐹) = (deg‘((Xp
∘f − (ℂ × {𝑧})) ∘f · 𝑄))) |
| 54 | 30 | peano2nnd 12283 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐷 + 1) ∈ ℕ) |
| 55 | 39, 54 | eqeltrrd 2842 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 56 | 55 | nnne0d 12316 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ≠ 0) |
| 57 | 40, 56 | eqnetrrid 3016 |
. . . . . . . . . . . 12
⊢ (𝜑 → (deg‘𝐹) ≠ 0) |
| 58 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝐹 = 0𝑝 →
(deg‘𝐹) =
(deg‘0𝑝)) |
| 59 | 58, 22 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝐹 = 0𝑝 →
(deg‘𝐹) =
0) |
| 60 | 59 | necon3i 2973 |
. . . . . . . . . . . 12
⊢
((deg‘𝐹) ≠
0 → 𝐹 ≠
0𝑝) |
| 61 | 57, 60 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ≠
0𝑝) |
| 62 | 61 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐹 ≠
0𝑝) |
| 63 | 52, 62 | eqnetrrd 3009 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → ((Xp
∘f − (ℂ × {𝑧})) ∘f · 𝑄) ≠
0𝑝) |
| 64 | | plymul0or 26322 |
. . . . . . . . . . 11
⊢
(((Xp ∘f − (ℂ ×
{𝑧})) ∈
(Poly‘ℂ) ∧ 𝑄 ∈ (Poly‘ℂ)) →
(((Xp ∘f − (ℂ × {𝑧})) ∘f ·
𝑄) = 0𝑝
↔ ((Xp ∘f − (ℂ ×
{𝑧})) =
0𝑝 ∨ 𝑄 = 0𝑝))) |
| 65 | 16, 28, 64 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (((Xp
∘f − (ℂ × {𝑧})) ∘f · 𝑄) = 0𝑝 ↔
((Xp ∘f − (ℂ × {𝑧})) = 0𝑝 ∨
𝑄 =
0𝑝))) |
| 66 | 65 | necon3abid 2977 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (((Xp
∘f − (ℂ × {𝑧})) ∘f · 𝑄) ≠ 0𝑝
↔ ¬ ((Xp ∘f − (ℂ
× {𝑧})) =
0𝑝 ∨ 𝑄 = 0𝑝))) |
| 67 | 63, 66 | mpbid 232 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → ¬ ((Xp
∘f − (ℂ × {𝑧})) = 0𝑝 ∨ 𝑄 =
0𝑝)) |
| 68 | | neanior 3035 |
. . . . . . . 8
⊢
(((Xp ∘f − (ℂ ×
{𝑧})) ≠
0𝑝 ∧ 𝑄 ≠ 0𝑝) ↔ ¬
((Xp ∘f − (ℂ × {𝑧})) = 0𝑝 ∨
𝑄 =
0𝑝)) |
| 69 | 67, 68 | sylibr 234 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → ((Xp
∘f − (ℂ × {𝑧})) ≠ 0𝑝 ∧ 𝑄 ≠
0𝑝)) |
| 70 | 69 | simprd 495 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝑄 ≠
0𝑝) |
| 71 | | eqid 2737 |
. . . . . . 7
⊢
(deg‘(Xp ∘f − (ℂ
× {𝑧}))) =
(deg‘(Xp ∘f − (ℂ ×
{𝑧}))) |
| 72 | | eqid 2737 |
. . . . . . 7
⊢
(deg‘𝑄) =
(deg‘𝑄) |
| 73 | 71, 72 | dgrmul 26310 |
. . . . . 6
⊢
((((Xp ∘f − (ℂ ×
{𝑧})) ∈
(Poly‘ℂ) ∧ (Xp ∘f −
(ℂ × {𝑧})) ≠
0𝑝) ∧ (𝑄 ∈ (Poly‘ℂ) ∧ 𝑄 ≠ 0𝑝))
→ (deg‘((Xp ∘f − (ℂ
× {𝑧}))
∘f · 𝑄)) = ((deg‘(Xp
∘f − (ℂ × {𝑧}))) + (deg‘𝑄))) |
| 74 | 16, 25, 28, 70, 73 | syl22anc 839 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (deg‘((Xp
∘f − (ℂ × {𝑧})) ∘f · 𝑄)) =
((deg‘(Xp ∘f − (ℂ ×
{𝑧}))) + (deg‘𝑄))) |
| 75 | 42, 53, 74 | 3eqtrd 2781 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝐷 + 1) = ((deg‘(Xp
∘f − (ℂ × {𝑧}))) + (deg‘𝑄))) |
| 76 | 17 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → ((deg‘(Xp
∘f − (ℂ × {𝑧}))) + (deg‘𝑄)) = (1 + (deg‘𝑄))) |
| 77 | 38, 75, 76 | 3eqtrd 2781 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (1 + 𝐷) = (1 + (deg‘𝑄))) |
| 78 | 29, 32, 35, 77 | addcanad 11466 |
. 2
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → 𝐷 = (deg‘𝑄)) |
| 79 | 28, 78 | jca 511 |
1
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄))) |