Proof of Theorem ply1termlem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ply1term.1 | . 2
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))) | 
| 2 |  | simplr 768 | . . . . . . 7
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ 𝑁 ∈
ℕ0) | 
| 3 |  | nn0uz 12921 | . . . . . . 7
⊢
ℕ0 = (ℤ≥‘0) | 
| 4 | 2, 3 | eleqtrdi 2850 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ 𝑁 ∈
(ℤ≥‘0)) | 
| 5 |  | fzss1 13604 | . . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑁...𝑁) ⊆ (0...𝑁)) | 
| 6 | 4, 5 | syl 17 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ (𝑁...𝑁) ⊆ (0...𝑁)) | 
| 7 |  | elfz1eq 13576 | . . . . . . . . 9
⊢ (𝑘 ∈ (𝑁...𝑁) → 𝑘 = 𝑁) | 
| 8 | 7 | adantl 481 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑘 = 𝑁) | 
| 9 |  | iftrue 4530 | . . . . . . . 8
⊢ (𝑘 = 𝑁 → if(𝑘 = 𝑁, 𝐴, 0) = 𝐴) | 
| 10 | 8, 9 | syl 17 | . . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → if(𝑘 = 𝑁, 𝐴, 0) = 𝐴) | 
| 11 |  | simpll 766 | . . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ 𝐴 ∈
ℂ) | 
| 12 | 11 | adantr 480 | . . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → 𝐴 ∈ ℂ) | 
| 13 | 10, 12 | eqeltrd 2840 | . . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ) | 
| 14 |  | simplr 768 | . . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑧 ∈ ℂ) | 
| 15 | 2 | adantr 480 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑁 ∈
ℕ0) | 
| 16 | 8, 15 | eqeltrd 2840 | . . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → 𝑘 ∈ ℕ0) | 
| 17 | 14, 16 | expcld 14187 | . . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → (𝑧↑𝑘) ∈ ℂ) | 
| 18 | 13, 17 | mulcld 11282 | . . . . 5
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ (𝑁...𝑁)) → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) ∈ ℂ) | 
| 19 |  | eldifn 4131 | . . . . . . . . . 10
⊢ (𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁)) → ¬ 𝑘 ∈ (𝑁...𝑁)) | 
| 20 | 19 | adantl 481 | . . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → ¬ 𝑘 ∈ (𝑁...𝑁)) | 
| 21 | 2 | adantr 480 | . . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → 𝑁 ∈
ℕ0) | 
| 22 | 21 | nn0zd 12641 | . . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → 𝑁 ∈ ℤ) | 
| 23 |  | fzsn 13607 | . . . . . . . . . . . 12
⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) | 
| 24 | 23 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 ∈ {𝑁})) | 
| 25 |  | elsn2g 4663 | . . . . . . . . . . 11
⊢ (𝑁 ∈ ℤ → (𝑘 ∈ {𝑁} ↔ 𝑘 = 𝑁)) | 
| 26 | 24, 25 | bitrd 279 | . . . . . . . . . 10
⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁)) | 
| 27 | 22, 26 | syl 17 | . . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁)) | 
| 28 | 20, 27 | mtbid 324 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → ¬ 𝑘 = 𝑁) | 
| 29 | 28 | iffalsed 4535 | . . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → if(𝑘 = 𝑁, 𝐴, 0) = 0) | 
| 30 | 29 | oveq1d 7447 | . . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘))) | 
| 31 |  | simpr 484 | . . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ 𝑧 ∈
ℂ) | 
| 32 |  | eldifi 4130 | . . . . . . . . 9
⊢ (𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁)) → 𝑘 ∈ (0...𝑁)) | 
| 33 |  | elfznn0 13661 | . . . . . . . . 9
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | 
| 34 | 32, 33 | syl 17 | . . . . . . . 8
⊢ (𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁)) → 𝑘 ∈ ℕ0) | 
| 35 |  | expcl 14121 | . . . . . . . 8
⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝑧↑𝑘) ∈
ℂ) | 
| 36 | 31, 34, 35 | syl2an 596 | . . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (𝑧↑𝑘) ∈ ℂ) | 
| 37 | 36 | mul02d 11460 | . . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (0 · (𝑧↑𝑘)) = 0) | 
| 38 | 30, 37 | eqtrd 2776 | . . . . 5
⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
∧ 𝑘 ∈ ((0...𝑁) ∖ (𝑁...𝑁))) → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = 0) | 
| 39 |  | fzfid 14015 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ (0...𝑁) ∈
Fin) | 
| 40 | 6, 18, 38, 39 | fsumss 15762 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ Σ𝑘 ∈
(𝑁...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))) | 
| 41 | 2 | nn0zd 12641 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ 𝑁 ∈
ℤ) | 
| 42 | 31, 2 | expcld 14187 | . . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ (𝑧↑𝑁) ∈
ℂ) | 
| 43 | 11, 42 | mulcld 11282 | . . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ (𝐴 · (𝑧↑𝑁)) ∈ ℂ) | 
| 44 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑘 = 𝑁 → (𝑧↑𝑘) = (𝑧↑𝑁)) | 
| 45 | 9, 44 | oveq12d 7450 | . . . . . 6
⊢ (𝑘 = 𝑁 → (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁))) | 
| 46 | 45 | fsum1 15784 | . . . . 5
⊢ ((𝑁 ∈ ℤ ∧ (𝐴 · (𝑧↑𝑁)) ∈ ℂ) → Σ𝑘 ∈ (𝑁...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁))) | 
| 47 | 41, 43, 46 | syl2anc 584 | . . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ Σ𝑘 ∈
(𝑁...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁))) | 
| 48 | 40, 47 | eqtr3d 2778 | . . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑧 ∈ ℂ)
→ Σ𝑘 ∈
(0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑁))) | 
| 49 | 48 | mpteq2dva 5241 | . 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (𝑧 ∈ ℂ
↦ Σ𝑘 ∈
(0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))) | 
| 50 | 1, 49 | eqtr4id 2795 | 1
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ 𝐹 = (𝑧 ∈ ℂ ↦
Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)))) |