Step | Hyp | Ref
| Expression |
1 | | binomcxplem.s |
. . . 4
⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
2 | | binomcxplem.p |
. . . . 5
⊢ 𝑃 = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘)) |
3 | | binomcxplem.d |
. . . . . . 7
⊢ 𝐷 = (◡abs “ (0[,)𝑅)) |
4 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑏◡abs |
5 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑏0 |
6 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑏[,) |
7 | | binomcxplem.r |
. . . . . . . . . 10
⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*,
< ) |
8 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑏
+ |
9 | | nfmpt1 4675 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑏(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
10 | 1, 9 | nfcxfr 2749 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑏𝑆 |
11 | | nfcv 2751 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑏𝑟 |
12 | 10, 11 | nffv 6110 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑏(𝑆‘𝑟) |
13 | 5, 8, 12 | nfseq 12673 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑏seq0(
+ , (𝑆‘𝑟)) |
14 | 13 | nfel1 2765 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑏seq0( + ,
(𝑆‘𝑟)) ∈ dom ⇝ |
15 | | nfcv 2751 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑏ℝ |
16 | 14, 15 | nfrab 3100 |
. . . . . . . . . . 11
⊢
Ⅎ𝑏{𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } |
17 | | nfcv 2751 |
. . . . . . . . . . 11
⊢
Ⅎ𝑏ℝ* |
18 | | nfcv 2751 |
. . . . . . . . . . 11
⊢
Ⅎ𝑏
< |
19 | 16, 17, 18 | nfsup 8240 |
. . . . . . . . . 10
⊢
Ⅎ𝑏sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*,
< ) |
20 | 7, 19 | nfcxfr 2749 |
. . . . . . . . 9
⊢
Ⅎ𝑏𝑅 |
21 | 5, 6, 20 | nfov 6575 |
. . . . . . . 8
⊢
Ⅎ𝑏(0[,)𝑅) |
22 | 4, 21 | nfima 5393 |
. . . . . . 7
⊢
Ⅎ𝑏(◡abs
“ (0[,)𝑅)) |
23 | 3, 22 | nfcxfr 2749 |
. . . . . 6
⊢
Ⅎ𝑏𝐷 |
24 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑦𝐷 |
25 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑦Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘) |
26 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑏ℕ0 |
27 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑏𝑦 |
28 | 10, 27 | nffv 6110 |
. . . . . . . 8
⊢
Ⅎ𝑏(𝑆‘𝑦) |
29 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑏𝑚 |
30 | 28, 29 | nffv 6110 |
. . . . . . 7
⊢
Ⅎ𝑏((𝑆‘𝑦)‘𝑚) |
31 | 26, 30 | nfsum 14269 |
. . . . . 6
⊢
Ⅎ𝑏Σ𝑚 ∈ ℕ0 ((𝑆‘𝑦)‘𝑚) |
32 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝑏 = 𝑦 ∧ 𝑘 ∈ ℕ0) → 𝑏 = 𝑦) |
33 | 32 | fveq2d 6107 |
. . . . . . . . 9
⊢ ((𝑏 = 𝑦 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑏) = (𝑆‘𝑦)) |
34 | 33 | fveq1d 6105 |
. . . . . . . 8
⊢ ((𝑏 = 𝑦 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑏)‘𝑘) = ((𝑆‘𝑦)‘𝑘)) |
35 | 34 | sumeq2dv 14281 |
. . . . . . 7
⊢ (𝑏 = 𝑦 → Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘) = Σ𝑘 ∈ ℕ0 ((𝑆‘𝑦)‘𝑘)) |
36 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑚((𝑆‘𝑦)‘𝑘) |
37 | | nfcv 2751 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘ℂ |
38 | | nfmpt1 4675 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘(𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))) |
39 | 37, 38 | nfmpt 4674 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
40 | 1, 39 | nfcxfr 2749 |
. . . . . . . . . 10
⊢
Ⅎ𝑘𝑆 |
41 | | nfcv 2751 |
. . . . . . . . . 10
⊢
Ⅎ𝑘𝑦 |
42 | 40, 41 | nffv 6110 |
. . . . . . . . 9
⊢
Ⅎ𝑘(𝑆‘𝑦) |
43 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑘𝑚 |
44 | 42, 43 | nffv 6110 |
. . . . . . . 8
⊢
Ⅎ𝑘((𝑆‘𝑦)‘𝑚) |
45 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → ((𝑆‘𝑦)‘𝑘) = ((𝑆‘𝑦)‘𝑚)) |
46 | 36, 44, 45 | cbvsumi 14275 |
. . . . . . 7
⊢
Σ𝑘 ∈
ℕ0 ((𝑆‘𝑦)‘𝑘) = Σ𝑚 ∈ ℕ0 ((𝑆‘𝑦)‘𝑚) |
47 | 35, 46 | syl6eq 2660 |
. . . . . 6
⊢ (𝑏 = 𝑦 → Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘) = Σ𝑚 ∈ ℕ0 ((𝑆‘𝑦)‘𝑚)) |
48 | 23, 24, 25, 31, 47 | cbvmptf 4676 |
. . . . 5
⊢ (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘)) = (𝑦 ∈ 𝐷 ↦ Σ𝑚 ∈ ℕ0 ((𝑆‘𝑦)‘𝑚)) |
49 | 2, 48 | eqtri 2632 |
. . . 4
⊢ 𝑃 = (𝑦 ∈ 𝐷 ↦ Σ𝑚 ∈ ℕ0 ((𝑆‘𝑦)‘𝑚)) |
50 | | ovex 6577 |
. . . . . 6
⊢ (𝐶C𝑐𝑗) ∈ V |
51 | 50 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐶C𝑐𝑗) ∈ V) |
52 | | binomcxplem.f |
. . . . . 6
⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) |
53 | 52 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))) |
54 | 52 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))) |
55 | | simpr 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 = 𝑘) → 𝑗 = 𝑘) |
56 | 55 | oveq2d 6565 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 = 𝑘) → (𝐶C𝑐𝑗) = (𝐶C𝑐𝑘)) |
57 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈
ℕ0) |
58 | | binomcxp.c |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℂ) |
59 | 58 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈
ℂ) |
60 | 59, 57 | bcccl 37560 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐶C𝑐𝑘) ∈
ℂ) |
61 | 54, 56, 57, 60 | fvmptd 6197 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐶C𝑐𝑘)) |
62 | 61, 60 | eqeltrd 2688 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
63 | 51, 53, 62 | fmpt2d 6300 |
. . . 4
⊢ (𝜑 → 𝐹:ℕ0⟶ℂ) |
64 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑟ℝ |
65 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑧ℝ |
66 | | nfv 1830 |
. . . . . . 7
⊢
Ⅎ𝑧seq0( + ,
(𝑆‘𝑟)) ∈ dom ⇝ |
67 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑟0 |
68 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑟
+ |
69 | | nfcv 2751 |
. . . . . . . . . . 11
⊢
Ⅎ𝑟(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
70 | 1, 69 | nfcxfr 2749 |
. . . . . . . . . 10
⊢
Ⅎ𝑟𝑆 |
71 | | nfcv 2751 |
. . . . . . . . . 10
⊢
Ⅎ𝑟𝑧 |
72 | 70, 71 | nffv 6110 |
. . . . . . . . 9
⊢
Ⅎ𝑟(𝑆‘𝑧) |
73 | 67, 68, 72 | nfseq 12673 |
. . . . . . . 8
⊢
Ⅎ𝑟seq0(
+ , (𝑆‘𝑧)) |
74 | 73 | nfel1 2765 |
. . . . . . 7
⊢
Ⅎ𝑟seq0( + ,
(𝑆‘𝑧)) ∈ dom ⇝ |
75 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑟 = 𝑧 → (𝑆‘𝑟) = (𝑆‘𝑧)) |
76 | 75 | seqeq3d 12671 |
. . . . . . . 8
⊢ (𝑟 = 𝑧 → seq0( + , (𝑆‘𝑟)) = seq0( + , (𝑆‘𝑧))) |
77 | 76 | eleq1d 2672 |
. . . . . . 7
⊢ (𝑟 = 𝑧 → (seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝑆‘𝑧)) ∈ dom ⇝ )) |
78 | 64, 65, 66, 74, 77 | cbvrab 3171 |
. . . . . 6
⊢ {𝑟 ∈ ℝ ∣ seq0( +
, (𝑆‘𝑟)) ∈ dom ⇝ } = {𝑧 ∈ ℝ ∣ seq0( +
, (𝑆‘𝑧)) ∈ dom ⇝
} |
79 | 78 | supeq1i 8236 |
. . . . 5
⊢
sup({𝑟 ∈
ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*,
< ) = sup({𝑧 ∈
ℝ ∣ seq0( + , (𝑆‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) |
80 | 7, 79 | eqtri 2632 |
. . . 4
⊢ 𝑅 = sup({𝑧 ∈ ℝ ∣ seq0( + , (𝑆‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) |
81 | 1 | fveq1i 6104 |
. . . . . . . . . . . 12
⊢ (𝑆‘𝑧) = ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧) |
82 | | seqeq3 12668 |
. . . . . . . . . . . 12
⊢ ((𝑆‘𝑧) = ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧) → seq0( + , (𝑆‘𝑧)) = seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧))) |
83 | 81, 82 | ax-mp 5 |
. . . . . . . . . . 11
⊢ seq0( + ,
(𝑆‘𝑧)) = seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) |
84 | 83 | eleq1i 2679 |
. . . . . . . . . 10
⊢ (seq0( +
, (𝑆‘𝑧)) ∈ dom ⇝ ↔
seq0( + , ((𝑏 ∈
ℂ ↦ (𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ ) |
85 | 84 | a1i 11 |
. . . . . . . . 9
⊢ (𝑧 ∈ ℝ → (seq0( +
, (𝑆‘𝑧)) ∈ dom ⇝ ↔
seq0( + , ((𝑏 ∈
ℂ ↦ (𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ )) |
86 | 85 | rabbiia 3161 |
. . . . . . . 8
⊢ {𝑧 ∈ ℝ ∣ seq0( +
, (𝑆‘𝑧)) ∈ dom ⇝ } = {𝑧 ∈ ℝ ∣ seq0( +
, ((𝑏 ∈ ℂ
↦ (𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ } |
87 | 86 | supeq1i 8236 |
. . . . . . 7
⊢
sup({𝑧 ∈
ℝ ∣ seq0( + , (𝑆‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) = sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) |
88 | 7, 79, 87 | 3eqtrri 2637 |
. . . . . 6
⊢
sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) = 𝑅 |
89 | 88 | eleq1i 2679 |
. . . . 5
⊢
(sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ ↔ 𝑅 ∈ ℝ) |
90 | 88 | oveq2i 6560 |
. . . . . 6
⊢
((abs‘𝑥) +
sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑏
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) = ((abs‘𝑥) +
𝑅) |
91 | 90 | oveq1i 6559 |
. . . . 5
⊢
(((abs‘𝑥) +
sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑏
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2) = (((abs‘𝑥) + 𝑅) / 2) |
92 | | eqid 2610 |
. . . . 5
⊢
((abs‘𝑥) + 1)
= ((abs‘𝑥) +
1) |
93 | 89, 91, 92 | ifbieq12i 4062 |
. . . 4
⊢
if(sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1)) = if(𝑅 ∈ ℝ, (((abs‘𝑥) + 𝑅) / 2), ((abs‘𝑥) + 1)) |
94 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = 𝑏 → (𝑤↑𝑘) = (𝑏↑𝑘)) |
95 | 94 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 𝑏 → ((𝐹‘𝑘) · (𝑤↑𝑘)) = ((𝐹‘𝑘) · (𝑏↑𝑘))) |
96 | 95 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑏 → (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))) = (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
97 | 96 | cbvmptv 4678 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘)))) = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
98 | 97 | fveq1i 6104 |
. . . . . . . . . . . . . 14
⊢ ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧) = ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧) |
99 | | seqeq3 12668 |
. . . . . . . . . . . . . 14
⊢ (((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧) = ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧) → seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) = seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧))) |
100 | 98, 99 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ seq0( + ,
((𝑤 ∈ ℂ ↦
(𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) = seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) |
101 | 100 | eleq1i 2679 |
. . . . . . . . . . . 12
⊢ (seq0( +
, ((𝑤 ∈ ℂ
↦ (𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ ↔ seq0( + ,
((𝑏 ∈ ℂ ↦
(𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ ) |
102 | 101 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ℝ → (seq0( +
, ((𝑤 ∈ ℂ
↦ (𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ ↔ seq0( + ,
((𝑏 ∈ ℂ ↦
(𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ )) |
103 | 102 | rabbiia 3161 |
. . . . . . . . . 10
⊢ {𝑧 ∈ ℝ ∣ seq0( +
, ((𝑤 ∈ ℂ
↦ (𝑘 ∈
ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ } = {𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ } |
104 | 103 | supeq1i 8236 |
. . . . . . . . 9
⊢
sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) = sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) |
105 | 104 | eleq1i 2679 |
. . . . . . . 8
⊢
(sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ ↔ sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ) |
106 | 104 | oveq2i 6560 |
. . . . . . . . 9
⊢
((abs‘𝑥) +
sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑤
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) = ((abs‘𝑥) +
sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑏
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) |
107 | 106 | oveq1i 6559 |
. . . . . . . 8
⊢
(((abs‘𝑥) +
sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑤
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2) = (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2) |
108 | 105, 107,
92 | ifbieq12i 4062 |
. . . . . . 7
⊢
if(sup({𝑧 ∈
ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1)) = if(sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1)) |
109 | 108 | oveq2i 6560 |
. . . . . 6
⊢
((abs‘𝑥) +
if(sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑤
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1))) = ((abs‘𝑥) + if(sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1))) |
110 | 109 | oveq1i 6559 |
. . . . 5
⊢
(((abs‘𝑥) +
if(sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑤
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1))) / 2) = (((abs‘𝑥) + if(sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1))) / 2) |
111 | 110 | oveq2i 6560 |
. . . 4
⊢
(0(ball‘(abs ∘ − ))(((abs‘𝑥) + if(sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑤 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑤↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1))) / 2)) = (0(ball‘(abs ∘
− ))(((abs‘𝑥) +
if(sup({𝑧 ∈ ℝ
∣ seq0( + , ((𝑏
∈ ℂ ↦ (𝑘
∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< ) ∈ ℝ, (((abs‘𝑥) + sup({𝑧 ∈ ℝ ∣ seq0( + , ((𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0
↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))‘𝑧)) ∈ dom ⇝ }, ℝ*,
< )) / 2), ((abs‘𝑥) + 1))) / 2)) |
112 | 1, 49, 63, 80, 3, 93, 111 | pserdv2 23988 |
. . 3
⊢ (𝜑 → (ℂ D 𝑃) = (𝑦 ∈ 𝐷 ↦ Σ𝑛 ∈ ℕ ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1))))) |
113 | | cnvimass 5404 |
. . . . . . . 8
⊢ (◡abs “ (0[,)𝑅)) ⊆ dom abs |
114 | 3, 113 | eqsstri 3598 |
. . . . . . 7
⊢ 𝐷 ⊆ dom
abs |
115 | | absf 13925 |
. . . . . . . 8
⊢
abs:ℂ⟶ℝ |
116 | 115 | fdmi 5965 |
. . . . . . 7
⊢ dom abs =
ℂ |
117 | 114, 116 | sseqtri 3600 |
. . . . . 6
⊢ 𝐷 ⊆
ℂ |
118 | 117 | sseli 3564 |
. . . . 5
⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ ℂ) |
119 | | binomcxplem.e |
. . . . . . . . . 10
⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))))) |
120 | 119 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))) |
121 | | simplr 788 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑏 = 𝑦) ∧ 𝑘 ∈ ℕ) → 𝑏 = 𝑦) |
122 | 121 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑏 = 𝑦) ∧ 𝑘 ∈ ℕ) → (𝑏↑(𝑘 − 1)) = (𝑦↑(𝑘 − 1))) |
123 | 122 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑏 = 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))) = ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1)))) |
124 | 123 | mpteq2dva 4672 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑏 = 𝑦) → (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1))))) |
125 | | simpr 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) |
126 | | nnex 10903 |
. . . . . . . . . . 11
⊢ ℕ
∈ V |
127 | 126 | mptex 6390 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1)))) ∈ V |
128 | 127 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1)))) ∈ V) |
129 | 120, 124,
125, 128 | fvmptd 6197 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐸‘𝑦) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1))))) |
130 | 129 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑦) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1))))) |
131 | | simpr 476 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → 𝑘 = 𝑛) |
132 | 131 | fveq2d 6107 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑘) = (𝐹‘𝑛)) |
133 | 131, 132 | oveq12d 6567 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝑘 · (𝐹‘𝑘)) = (𝑛 · (𝐹‘𝑛))) |
134 | 131 | oveq1d 6564 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝑘 − 1) = (𝑛 − 1)) |
135 | 134 | oveq2d 6565 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝑦↑(𝑘 − 1)) = (𝑦↑(𝑛 − 1))) |
136 | 133, 135 | oveq12d 6567 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ((𝑘 · (𝐹‘𝑘)) · (𝑦↑(𝑘 − 1))) = ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1)))) |
137 | | simpr 476 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
138 | | ovex 6577 |
. . . . . . . 8
⊢ ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V |
139 | 138 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) → ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V) |
140 | 130, 136,
137, 139 | fvmptd 6197 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℂ) ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑦)‘𝑛) = ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1)))) |
141 | 140 | sumeq2dv 14281 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛) = Σ𝑛 ∈ ℕ ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1)))) |
142 | 118, 141 | sylan2 490 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛) = Σ𝑛 ∈ ℕ ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1)))) |
143 | 142 | mpteq2dva 4672 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛)) = (𝑦 ∈ 𝐷 ↦ Σ𝑛 ∈ ℕ ((𝑛 · (𝐹‘𝑛)) · (𝑦↑(𝑛 − 1))))) |
144 | 112, 143 | eqtr4d 2647 |
. 2
⊢ (𝜑 → (ℂ D 𝑃) = (𝑦 ∈ 𝐷 ↦ Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛))) |
145 | | nfcv 2751 |
. . . 4
⊢
Ⅎ𝑏ℕ |
146 | | nfmpt1 4675 |
. . . . . . 7
⊢
Ⅎ𝑏(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))))) |
147 | 119, 146 | nfcxfr 2749 |
. . . . . 6
⊢
Ⅎ𝑏𝐸 |
148 | 147, 27 | nffv 6110 |
. . . . 5
⊢
Ⅎ𝑏(𝐸‘𝑦) |
149 | | nfcv 2751 |
. . . . 5
⊢
Ⅎ𝑏𝑛 |
150 | 148, 149 | nffv 6110 |
. . . 4
⊢
Ⅎ𝑏((𝐸‘𝑦)‘𝑛) |
151 | 145, 150 | nfsum 14269 |
. . 3
⊢
Ⅎ𝑏Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛) |
152 | | nfcv 2751 |
. . 3
⊢
Ⅎ𝑦Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘) |
153 | | simpl 472 |
. . . . . . 7
⊢ ((𝑦 = 𝑏 ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑏) |
154 | 153 | fveq2d 6107 |
. . . . . 6
⊢ ((𝑦 = 𝑏 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑦) = (𝐸‘𝑏)) |
155 | 154 | fveq1d 6105 |
. . . . 5
⊢ ((𝑦 = 𝑏 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑦)‘𝑛) = ((𝐸‘𝑏)‘𝑛)) |
156 | 155 | sumeq2dv 14281 |
. . . 4
⊢ (𝑦 = 𝑏 → Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛) = Σ𝑛 ∈ ℕ ((𝐸‘𝑏)‘𝑛)) |
157 | | nfmpt1 4675 |
. . . . . . . . 9
⊢
Ⅎ𝑘(𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))) |
158 | 37, 157 | nfmpt 4674 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))))) |
159 | 119, 158 | nfcxfr 2749 |
. . . . . . 7
⊢
Ⅎ𝑘𝐸 |
160 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑘𝑏 |
161 | 159, 160 | nffv 6110 |
. . . . . 6
⊢
Ⅎ𝑘(𝐸‘𝑏) |
162 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑘𝑛 |
163 | 161, 162 | nffv 6110 |
. . . . 5
⊢
Ⅎ𝑘((𝐸‘𝑏)‘𝑛) |
164 | | nfcv 2751 |
. . . . 5
⊢
Ⅎ𝑛((𝐸‘𝑏)‘𝑘) |
165 | | fveq2 6103 |
. . . . 5
⊢ (𝑛 = 𝑘 → ((𝐸‘𝑏)‘𝑛) = ((𝐸‘𝑏)‘𝑘)) |
166 | 163, 164,
165 | cbvsumi 14275 |
. . . 4
⊢
Σ𝑛 ∈
ℕ ((𝐸‘𝑏)‘𝑛) = Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘) |
167 | 156, 166 | syl6eq 2660 |
. . 3
⊢ (𝑦 = 𝑏 → Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛) = Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘)) |
168 | 24, 23, 151, 152, 167 | cbvmptf 4676 |
. 2
⊢ (𝑦 ∈ 𝐷 ↦ Σ𝑛 ∈ ℕ ((𝐸‘𝑦)‘𝑛)) = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘)) |
169 | 144, 168 | syl6eq 2660 |
1
⊢ (𝜑 → (ℂ D 𝑃) = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘))) |