| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | knoppcnlem10.f | . . . 4
⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | 
| 2 |  | simpr 484 | . . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ) | 
| 3 |  | knoppcnlem10.2 | . . . . 5
⊢ (𝜑 → 𝑀 ∈
ℕ0) | 
| 4 | 3 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 𝑀 ∈
ℕ0) | 
| 5 | 1, 2, 4 | knoppcnlem1 36494 | . . 3
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝑧)))) | 
| 6 | 5 | mpteq2dva 5242 | . 2
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) = (𝑧 ∈ ℝ ↦ ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝑧))))) | 
| 7 |  | retopon 24784 | . . . 4
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) | 
| 8 | 7 | a1i 11 | . . 3
⊢ (𝜑 → (topGen‘ran (,))
∈ (TopOn‘ℝ)) | 
| 9 |  | eqid 2737 | . . . . . 6
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) | 
| 10 | 9 | cnfldtopon 24803 | . . . . 5
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) | 
| 11 | 10 | a1i 11 | . . . 4
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) | 
| 12 |  | knoppcnlem10.1 | . . . . . 6
⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| 13 | 12 | recnd 11289 | . . . . 5
⊢ (𝜑 → 𝐶 ∈ ℂ) | 
| 14 | 13, 3 | expcld 14186 | . . . 4
⊢ (𝜑 → (𝐶↑𝑀) ∈ ℂ) | 
| 15 | 8, 11, 14 | cnmptc 23670 | . . 3
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ (𝐶↑𝑀)) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 16 |  | 2cnd 12344 | . . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℂ) | 
| 17 |  | knoppcnlem10.n | . . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 18 | 17 | nncnd 12282 | . . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℂ) | 
| 19 | 16, 18 | mulcld 11281 | . . . . . . . . 9
⊢ (𝜑 → (2 · 𝑁) ∈
ℂ) | 
| 20 | 19, 3 | expcld 14186 | . . . . . . . 8
⊢ (𝜑 → ((2 · 𝑁)↑𝑀) ∈ ℂ) | 
| 21 | 8, 11, 20 | cnmptc 23670 | . . . . . . 7
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((2 · 𝑁)↑𝑀)) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 22 |  | tgioo4 24826 | . . . . . . . . . 10
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) | 
| 23 | 22 | oveq2i 7442 | . . . . . . . . 9
⊢
((topGen‘ran (,)) Cn (topGen‘ran (,))) = ((topGen‘ran
(,)) Cn ((TopOpen‘ℂfld) ↾t
ℝ)) | 
| 24 | 9 | cnfldtop 24804 | . . . . . . . . . 10
⊢
(TopOpen‘ℂfld) ∈ Top | 
| 25 |  | cnrest2r 23295 | . . . . . . . . . 10
⊢
((TopOpen‘ℂfld) ∈ Top →
((topGen‘ran (,)) Cn ((TopOpen‘ℂfld)
↾t ℝ)) ⊆ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 26 | 24, 25 | ax-mp 5 | . . . . . . . . 9
⊢
((topGen‘ran (,)) Cn ((TopOpen‘ℂfld)
↾t ℝ)) ⊆ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld)) | 
| 27 | 23, 26 | eqsstri 4030 | . . . . . . . 8
⊢
((topGen‘ran (,)) Cn (topGen‘ran (,))) ⊆
((topGen‘ran (,)) Cn
(TopOpen‘ℂfld)) | 
| 28 | 8 | cnmptid 23669 | . . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 𝑧) ∈ ((topGen‘ran (,)) Cn
(topGen‘ran (,)))) | 
| 29 | 27, 28 | sselid 3981 | . . . . . . 7
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ 𝑧) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 30 | 9 | mpomulcn 24891 | . . . . . . . 8
⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) | 
| 31 | 30 | a1i 11 | . . . . . . 7
⊢ (𝜑 → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈
(((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) | 
| 32 |  | oveq12 7440 | . . . . . . 7
⊢ ((𝑢 = ((2 · 𝑁)↑𝑀) ∧ 𝑣 = 𝑧) → (𝑢 · 𝑣) = (((2 · 𝑁)↑𝑀) · 𝑧)) | 
| 33 | 8, 21, 29, 11, 11, 31, 32 | cnmpt12 23675 | . . . . . 6
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 34 |  | 2re 12340 | . . . . . . . . . . . . . 14
⊢ 2 ∈
ℝ | 
| 35 | 34 | a1i 11 | . . . . . . . . . . . . 13
⊢ (𝜑 → 2 ∈
ℝ) | 
| 36 | 17 | nnred 12281 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℝ) | 
| 37 | 35, 36 | remulcld 11291 | . . . . . . . . . . . 12
⊢ (𝜑 → (2 · 𝑁) ∈
ℝ) | 
| 38 | 37, 3 | reexpcld 14203 | . . . . . . . . . . 11
⊢ (𝜑 → ((2 · 𝑁)↑𝑀) ∈ ℝ) | 
| 39 | 38 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((2 · 𝑁)↑𝑀) ∈ ℝ) | 
| 40 | 39, 2 | remulcld 11291 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (((2 · 𝑁)↑𝑀) · 𝑧) ∈ ℝ) | 
| 41 | 40 | fmpttd 7135 | . . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)):ℝ⟶ℝ) | 
| 42 | 41 | frnd 6744 | . . . . . . 7
⊢ (𝜑 → ran (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ⊆ ℝ) | 
| 43 |  | ax-resscn 11212 | . . . . . . . 8
⊢ ℝ
⊆ ℂ | 
| 44 | 43 | a1i 11 | . . . . . . 7
⊢ (𝜑 → ℝ ⊆
ℂ) | 
| 45 |  | cnrest2 23294 | . . . . . . 7
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ ran (𝑧 ∈ ℝ
↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ⊆ ℝ ∧ ℝ ⊆
ℂ) → ((𝑧 ∈
ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld)) ↔ (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
((TopOpen‘ℂfld) ↾t
ℝ)))) | 
| 46 | 10, 42, 44, 45 | mp3an2i 1468 | . . . . . 6
⊢ (𝜑 → ((𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld)) ↔ (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
((TopOpen‘ℂfld) ↾t
ℝ)))) | 
| 47 | 33, 46 | mpbid 232 | . . . . 5
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
((TopOpen‘ℂfld) ↾t
ℝ))) | 
| 48 | 47, 23 | eleqtrrdi 2852 | . . . 4
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ (((2 · 𝑁)↑𝑀) · 𝑧)) ∈ ((topGen‘ran (,)) Cn
(topGen‘ran (,)))) | 
| 49 |  | ssid 4006 | . . . . . . 7
⊢ ℂ
⊆ ℂ | 
| 50 |  | cncfss 24925 | . . . . . . 7
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)) | 
| 51 | 43, 49, 50 | mp2an 692 | . . . . . 6
⊢
(ℝ–cn→ℝ)
⊆ (ℝ–cn→ℂ) | 
| 52 |  | knoppcnlem10.t | . . . . . . . 8
⊢ 𝑇 = (𝑥 ∈ ℝ ↦
(abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | 
| 53 | 52 | dnicn 36493 | . . . . . . 7
⊢ 𝑇 ∈ (ℝ–cn→ℝ) | 
| 54 | 53 | a1i 11 | . . . . . 6
⊢ (𝜑 → 𝑇 ∈ (ℝ–cn→ℝ)) | 
| 55 | 51, 54 | sselid 3981 | . . . . 5
⊢ (𝜑 → 𝑇 ∈ (ℝ–cn→ℂ)) | 
| 56 | 10 | toponrestid 22927 | . . . . . . 7
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) | 
| 57 | 9, 22, 56 | cncfcn 24936 | . . . . . 6
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℂ) = ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 58 | 43, 49, 57 | mp2an 692 | . . . . 5
⊢
(ℝ–cn→ℂ) =
((topGen‘ran (,)) Cn
(TopOpen‘ℂfld)) | 
| 59 | 55, 58 | eleqtrdi 2851 | . . . 4
⊢ (𝜑 → 𝑇 ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 60 | 8, 48, 59 | cnmpt11f 23672 | . . 3
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ (𝑇‘(((2 · 𝑁)↑𝑀) · 𝑧))) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 61 |  | oveq12 7440 | . . 3
⊢ ((𝑢 = (𝐶↑𝑀) ∧ 𝑣 = (𝑇‘(((2 · 𝑁)↑𝑀) · 𝑧))) → (𝑢 · 𝑣) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝑧)))) | 
| 62 | 8, 15, 60, 11, 11, 31, 61 | cnmpt12 23675 | . 2
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝑧)))) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) | 
| 63 | 6, 62 | eqeltrd 2841 | 1
⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn
(TopOpen‘ℂfld))) |