Step | Hyp | Ref
| Expression |
1 | | nnuz 11723 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
2 | | 1zzd 11408 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℤ) |
3 | | fzfid 12772 |
. . . . . 6
⊢ (𝜑 → (0...𝑁) ∈ Fin) |
4 | | 1zzd 11408 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → 1 ∈ ℤ) |
5 | | plyeq0.3 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚
ℕ0)) |
6 | | plyeq0.1 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
7 | | 0cn 10032 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ∈
ℂ |
8 | 7 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 ∈
ℂ) |
9 | 8 | snssd 4340 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → {0} ⊆
ℂ) |
10 | 6, 9 | unssd 3789 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑆 ∪ {0}) ⊆
ℂ) |
11 | | cnex 10017 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℂ
∈ V |
12 | | ssexg 4804 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑆 ∪ {0}) ⊆ ℂ
∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V) |
13 | 10, 11, 12 | sylancl 694 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑆 ∪ {0}) ∈ V) |
14 | | nn0ex 11298 |
. . . . . . . . . . . . . . . . . 18
⊢
ℕ0 ∈ V |
15 | | elmapg 7870 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑆 ∪ {0}) ∈ V ∧
ℕ0 ∈ V) → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚
ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) |
16 | 13, 14, 15 | sylancl 694 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚
ℕ0) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0}))) |
17 | 5, 16 | mpbid 222 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0})) |
18 | 17, 10 | fssd 6057 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
19 | | elfznn0 12433 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) |
20 | | ffvelrn 6357 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴:ℕ0⟶ℂ ∧
𝑘 ∈
ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
21 | 18, 19, 20 | syl2an 494 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
22 | 21 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝐴‘𝑘) ∈ ℂ) |
23 | 22 | abscld 14175 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (abs‘(𝐴‘𝑘)) ∈ ℝ) |
24 | 23 | recnd 10068 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (abs‘(𝐴‘𝑘)) ∈ ℂ) |
25 | | divcnv 14585 |
. . . . . . . . . . 11
⊢
((abs‘(𝐴‘𝑘)) ∈ ℂ → (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛)) ⇝ 0) |
26 | 24, 25 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛)) ⇝ 0) |
27 | | nnex 11026 |
. . . . . . . . . . . 12
⊢ ℕ
∈ V |
28 | 27 | mptex 6486 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦
((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀)))) ∈ V |
29 | 28 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀)))) ∈ V) |
30 | | oveq2 6658 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → ((abs‘(𝐴‘𝑘)) / 𝑛) = ((abs‘(𝐴‘𝑘)) / 𝑚)) |
31 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ ↦
((abs‘(𝐴‘𝑘)) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛)) |
32 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢
((abs‘(𝐴‘𝑘)) / 𝑚) ∈ V |
33 | 30, 31, 32 | fvmpt 6282 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦
((abs‘(𝐴‘𝑘)) / 𝑛))‘𝑚) = ((abs‘(𝐴‘𝑘)) / 𝑚)) |
34 | 33 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛))‘𝑚) = ((abs‘(𝐴‘𝑘)) / 𝑚)) |
35 | | nndivre 11056 |
. . . . . . . . . . . 12
⊢
(((abs‘(𝐴‘𝑘)) ∈ ℝ ∧ 𝑚 ∈ ℕ) → ((abs‘(𝐴‘𝑘)) / 𝑚) ∈ ℝ) |
36 | 23, 35 | sylan 488 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((abs‘(𝐴‘𝑘)) / 𝑚) ∈ ℝ) |
37 | 34, 36 | eqeltrd 2701 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛))‘𝑚) ∈ ℝ) |
38 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → (𝑛↑(𝑘 − 𝑀)) = (𝑚↑(𝑘 − 𝑀))) |
39 | 38 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))) = ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀)))) |
40 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ ↦
((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀)))) = (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀)))) |
41 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢
((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀))) ∈ V |
42 | 39, 40, 41 | fvmpt 6282 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦
((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀)))) |
43 | 42 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀)))) |
44 | 21 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝐴‘𝑘) ∈ ℂ) |
45 | 44 | abscld 14175 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (abs‘(𝐴‘𝑘)) ∈ ℝ) |
46 | | nnrp 11842 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℝ+) |
47 | 46 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ+) |
48 | | elfzelz 12342 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) |
49 | | cnvimass 5485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (◡𝐴 “ (𝑆 ∖ {0})) ⊆ dom 𝐴 |
50 | | fdm 6051 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → dom 𝐴 =
ℕ0) |
51 | 17, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → dom 𝐴 = ℕ0) |
52 | 49, 51 | syl5sseq 3653 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (◡𝐴 “ (𝑆 ∖ {0})) ⊆
ℕ0) |
53 | | plyeq0.6 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑀 = sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, <
) |
54 | | nn0ssz 11398 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
ℕ0 ⊆ ℤ |
55 | 52, 54 | syl6ss 3615 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (◡𝐴 “ (𝑆 ∖ {0})) ⊆
ℤ) |
56 | | plyeq0.7 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (◡𝐴 “ (𝑆 ∖ {0})) ≠
∅) |
57 | | plyeq0.2 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
58 | 57 | nn0red 11352 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ ℝ) |
59 | 52 | sselda 3603 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) → 𝑧 ∈ ℕ0) |
60 | | plyeq0.4 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
61 | | plyco0 23948 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑁 ∈ ℕ0
∧ 𝐴:ℕ0⟶ℂ) →
((𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0
((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
62 | 57, 18, 61 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((𝐴 “
(ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0
((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
63 | 60, 62 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
64 | 63 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) → ∀𝑘 ∈ ℕ0
((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
65 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn
ℕ0) |
66 | 17, 65 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝐴 Fn ℕ0) |
67 | | elpreima 6337 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝐴 Fn ℕ0 →
(𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑧 ∈ ℕ0 ∧ (𝐴‘𝑧) ∈ (𝑆 ∖ {0})))) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑧 ∈ ℕ0 ∧ (𝐴‘𝑧) ∈ (𝑆 ∖ {0})))) |
69 | 68 | simplbda 654 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) → (𝐴‘𝑧) ∈ (𝑆 ∖ {0})) |
70 | | eldifsni 4320 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴‘𝑧) ∈ (𝑆 ∖ {0}) → (𝐴‘𝑧) ≠ 0) |
71 | 69, 70 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) → (𝐴‘𝑧) ≠ 0) |
72 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 𝑧 → (𝐴‘𝑘) = (𝐴‘𝑧)) |
73 | 72 | neeq1d 2853 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 𝑧 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑧) ≠ 0)) |
74 | | breq1 4656 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 𝑧 → (𝑘 ≤ 𝑁 ↔ 𝑧 ≤ 𝑁)) |
75 | 73, 74 | imbi12d 334 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 𝑧 → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘𝑧) ≠ 0 → 𝑧 ≤ 𝑁))) |
76 | 75 | rspcv 3305 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 ∈ ℕ0
→ (∀𝑘 ∈
ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) → ((𝐴‘𝑧) ≠ 0 → 𝑧 ≤ 𝑁))) |
77 | 59, 64, 71, 76 | syl3c 66 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) → 𝑧 ≤ 𝑁) |
78 | 77 | ralrimiva 2966 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑁) |
79 | | breq2 4657 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑁 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝑁)) |
80 | 79 | ralbidv 2986 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑁 → (∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑁)) |
81 | 80 | rspcev 3309 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈ ℝ ∧
∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑁) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥) |
82 | 58, 78, 81 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥) |
83 | | suprzcl 11457 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((◡𝐴 “ (𝑆 ∖ {0})) ⊆ ℤ ∧ (◡𝐴 “ (𝑆 ∖ {0})) ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥) → sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) ∈
(◡𝐴 “ (𝑆 ∖ {0}))) |
84 | 55, 56, 82, 83 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) ∈
(◡𝐴 “ (𝑆 ∖ {0}))) |
85 | 53, 84 | syl5eqel 2705 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) |
86 | 52, 85 | sseldd 3604 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
87 | 86 | nn0zd 11480 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℤ) |
88 | | zsubcl 11419 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 − 𝑀) ∈ ℤ) |
89 | 48, 87, 88 | syl2anr 495 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 − 𝑀) ∈ ℤ) |
90 | 89 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑘 − 𝑀) ∈ ℤ) |
91 | 47, 90 | rpexpcld 13032 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑚↑(𝑘 − 𝑀)) ∈
ℝ+) |
92 | 91 | rpred 11872 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑚↑(𝑘 − 𝑀)) ∈ ℝ) |
93 | 45, 92 | remulcld 10070 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀))) ∈ ℝ) |
94 | 43, 93 | eqeltrd 2701 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) ∈ ℝ) |
95 | | nnrecre 11057 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → (1 /
𝑚) ∈
ℝ) |
96 | 95 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈
ℝ) |
97 | 22 | absge0d 14183 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → 0 ≤ (abs‘(𝐴‘𝑘))) |
98 | 97 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 0 ≤
(abs‘(𝐴‘𝑘))) |
99 | | nnre 11027 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℝ) |
100 | 99 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ) |
101 | | nnge1 11046 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 1 ≤
𝑚) |
102 | 101 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 1 ≤ 𝑚) |
103 | | 1red 10055 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 1 ∈
ℝ) |
104 | 90 | zred 11482 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑘 − 𝑀) ∈ ℝ) |
105 | | simplr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑘 < 𝑀) |
106 | 48 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ) |
107 | 106 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑘 ∈ ℤ) |
108 | 87 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ ℤ) |
109 | | zltp1le 11427 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 < 𝑀 ↔ (𝑘 + 1) ≤ 𝑀)) |
110 | 107, 108,
109 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑘 < 𝑀 ↔ (𝑘 + 1) ≤ 𝑀)) |
111 | 105, 110 | mpbid 222 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑘 + 1) ≤ 𝑀) |
112 | 19 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
113 | 112 | nn0red 11352 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ) |
114 | 113 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑘 ∈ ℝ) |
115 | 86 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑀 ∈
ℕ0) |
116 | 115 | nn0red 11352 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑀 ∈ ℝ) |
117 | 116 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ ℝ) |
118 | 114, 103,
117 | leaddsub2d 10629 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑘 + 1) ≤ 𝑀 ↔ 1 ≤ (𝑀 − 𝑘))) |
119 | 111, 118 | mpbid 222 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 1 ≤ (𝑀 − 𝑘)) |
120 | 113 | recnd 10068 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ) |
121 | 120 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑘 ∈ ℂ) |
122 | 116 | recnd 10068 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑀 ∈ ℂ) |
123 | 122 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ ℂ) |
124 | 121, 123 | negsubdi2d 10408 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → -(𝑘 − 𝑀) = (𝑀 − 𝑘)) |
125 | 119, 124 | breqtrrd 4681 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 1 ≤ -(𝑘 − 𝑀)) |
126 | 103, 104,
125 | lenegcon2d 10610 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑘 − 𝑀) ≤ -1) |
127 | | neg1z 11413 |
. . . . . . . . . . . . . . . 16
⊢ -1 ∈
ℤ |
128 | | eluz 11701 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 − 𝑀) ∈ ℤ ∧ -1 ∈ ℤ)
→ (-1 ∈ (ℤ≥‘(𝑘 − 𝑀)) ↔ (𝑘 − 𝑀) ≤ -1)) |
129 | 90, 127, 128 | sylancl 694 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (-1 ∈
(ℤ≥‘(𝑘 − 𝑀)) ↔ (𝑘 − 𝑀) ≤ -1)) |
130 | 126, 129 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → -1 ∈
(ℤ≥‘(𝑘 − 𝑀))) |
131 | 100, 102,
130 | leexp2ad 13041 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑚↑(𝑘 − 𝑀)) ≤ (𝑚↑-1)) |
132 | | nncn 11028 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
133 | 132 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
134 | | expn1 12870 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℂ → (𝑚↑-1) = (1 / 𝑚)) |
135 | 133, 134 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑚↑-1) = (1 / 𝑚)) |
136 | 131, 135 | breqtrd 4679 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑚↑(𝑘 − 𝑀)) ≤ (1 / 𝑚)) |
137 | 92, 96, 45, 98, 136 | lemul2ad 10964 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀))) ≤ ((abs‘(𝐴‘𝑘)) · (1 / 𝑚))) |
138 | 24 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (abs‘(𝐴‘𝑘)) ∈ ℂ) |
139 | | nnne0 11053 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) |
140 | 139 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
141 | 138, 133,
140 | divrecd 10804 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((abs‘(𝐴‘𝑘)) / 𝑚) = ((abs‘(𝐴‘𝑘)) · (1 / 𝑚))) |
142 | 34, 141 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛))‘𝑚) = ((abs‘(𝐴‘𝑘)) · (1 / 𝑚))) |
143 | 137, 43, 142 | 3brtr4d 4685 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) ≤ ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) / 𝑛))‘𝑚)) |
144 | 91 | rpge0d 11876 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 0 ≤ (𝑚↑(𝑘 − 𝑀))) |
145 | 45, 92, 98, 144 | mulge0d 10604 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 0 ≤
((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀)))) |
146 | 145, 43 | breqtrrd 4681 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ ↦
((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))))‘𝑚)) |
147 | 1, 4, 26, 29, 37, 94, 143, 146 | climsqz2 14372 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀)))) ⇝ 0) |
148 | 27 | mptex 6486 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ∈ V |
149 | 148 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ∈ V) |
150 | 38 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))) = ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀)))) |
151 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) = (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) |
152 | | ovex 6678 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀))) ∈ V |
153 | 150, 151,
152 | fvmpt 6282 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀)))) |
154 | 153 | ad2antlr 763 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀)))) |
155 | 18 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴:ℕ0⟶ℂ) |
156 | 155, 19, 20 | syl2an 494 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
157 | 132 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑚 ∈ ℂ) |
158 | 139 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑚 ≠ 0) |
159 | 87 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑀 ∈ ℤ) |
160 | 48, 159, 88 | syl2anr 495 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 − 𝑀) ∈ ℤ) |
161 | 157, 158,
160 | expclzd 13013 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑚↑(𝑘 − 𝑀)) ∈ ℂ) |
162 | 156, 161 | mulcld 10060 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀))) ∈ ℂ) |
163 | 154, 162 | eqeltrd 2701 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) ∈ ℂ) |
164 | 163 | an32s 846 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) ∈ ℂ) |
165 | 164 | adantlr 751 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) ∈ ℂ) |
166 | 92 | recnd 10068 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (𝑚↑(𝑘 − 𝑀)) ∈ ℂ) |
167 | 44, 166 | absmuld 14193 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (abs‘((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀)))) = ((abs‘(𝐴‘𝑘)) · (abs‘(𝑚↑(𝑘 − 𝑀))))) |
168 | 92, 144 | absidd 14161 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (abs‘(𝑚↑(𝑘 − 𝑀))) = (𝑚↑(𝑘 − 𝑀))) |
169 | 168 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((abs‘(𝐴‘𝑘)) · (abs‘(𝑚↑(𝑘 − 𝑀)))) = ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀)))) |
170 | 167, 169 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (abs‘((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀)))) = ((abs‘(𝐴‘𝑘)) · (𝑚↑(𝑘 − 𝑀)))) |
171 | 153 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀)))) |
172 | 171 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → (abs‘((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚)) = (abs‘((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀))))) |
173 | 170, 172,
43 | 3eqtr4rd 2667 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = (abs‘((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚))) |
174 | 1, 4, 149, 29, 165, 173 | climabs0 14316 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ ((abs‘(𝐴‘𝑘)) · (𝑛↑(𝑘 − 𝑀)))) ⇝ 0)) |
175 | 147, 174 | mpbird 247 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ⇝ 0) |
176 | 113 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → 𝑘 ∈ ℝ) |
177 | | simpr 477 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → 𝑘 < 𝑀) |
178 | 176, 177 | ltned 10173 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → 𝑘 ≠ 𝑀) |
179 | | velsn 4193 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ {𝑀} ↔ 𝑘 = 𝑀) |
180 | 179 | necon3bbii 2841 |
. . . . . . . . . 10
⊢ (¬
𝑘 ∈ {𝑀} ↔ 𝑘 ≠ 𝑀) |
181 | 178, 180 | sylibr 224 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → ¬ 𝑘 ∈ {𝑀}) |
182 | 181 | iffalsed 4097 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) = 0) |
183 | 175, 182 | breqtrrd 4681 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑘 < 𝑀) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ⇝ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
184 | | nncn 11028 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
185 | 184 | ad2antlr 763 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → 𝑛 ∈ ℂ) |
186 | | nnne0 11053 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
187 | 186 | ad2antlr 763 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → 𝑛 ≠ 0) |
188 | 89 | ad3antrrr 766 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → (𝑘 − 𝑀) ∈ ℤ) |
189 | 185, 187,
188 | expclzd 13013 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → (𝑛↑(𝑘 − 𝑀)) ∈ ℂ) |
190 | 189 | mul02d 10234 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → (0 · (𝑛↑(𝑘 − 𝑀))) = 0) |
191 | | simpr 477 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → (𝐴‘𝑘) = 0) |
192 | 191 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))) = (0 · (𝑛↑(𝑘 − 𝑀)))) |
193 | 191 | ifeq1d 4104 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) = if(𝑘 ∈ {𝑀}, 0, 0)) |
194 | | ifid 4125 |
. . . . . . . . . . . . 13
⊢ if(𝑘 ∈ {𝑀}, 0, 0) = 0 |
195 | 193, 194 | syl6eq 2672 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) = 0) |
196 | 190, 192,
195 | 3eqtr4d 2666 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) = 0) → ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))) = if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
197 | 21 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) → (𝐴‘𝑘) ∈ ℂ) |
198 | 197 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝐴‘𝑘) ∈ ℂ) |
199 | 198 | mulid1d 10057 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → ((𝐴‘𝑘) · 1) = (𝐴‘𝑘)) |
200 | | nn0ssre 11296 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
ℕ0 ⊆ ℝ |
201 | 52, 200 | syl6ss 3615 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (◡𝐴 “ (𝑆 ∖ {0})) ⊆
ℝ) |
202 | 201 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → (◡𝐴 “ (𝑆 ∖ {0})) ⊆
ℝ) |
203 | 56 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → (◡𝐴 “ (𝑆 ∖ {0})) ≠
∅) |
204 | 82 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥) |
205 | 19 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ∈ ℕ0) |
206 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0)
→ (𝐴‘𝑘) ∈ (𝑆 ∪ {0})) |
207 | 17, 19, 206 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ (𝑆 ∪ {0})) |
208 | 207 | anim1i 592 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → ((𝐴‘𝑘) ∈ (𝑆 ∪ {0}) ∧ (𝐴‘𝑘) ≠ 0)) |
209 | | eldifsn 4317 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐴‘𝑘) ∈ ((𝑆 ∪ {0}) ∖ {0}) ↔ ((𝐴‘𝑘) ∈ (𝑆 ∪ {0}) ∧ (𝐴‘𝑘) ≠ 0)) |
210 | 208, 209 | sylibr 224 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → (𝐴‘𝑘) ∈ ((𝑆 ∪ {0}) ∖ {0})) |
211 | | difun2 4048 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑆 ∪ {0}) ∖ {0}) =
(𝑆 ∖
{0}) |
212 | 210, 211 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → (𝐴‘𝑘) ∈ (𝑆 ∖ {0})) |
213 | | elpreima 6337 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐴 Fn ℕ0 →
(𝑘 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ∈ (𝑆 ∖ {0})))) |
214 | 66, 213 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑘 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ∈ (𝑆 ∖ {0})))) |
215 | 214 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → (𝑘 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ∈ (𝑆 ∖ {0})))) |
216 | 205, 212,
215 | mpbir2and 957 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) |
217 | | suprub 10984 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((◡𝐴 “ (𝑆 ∖ {0})) ⊆ ℝ ∧ (◡𝐴 “ (𝑆 ∖ {0})) ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥) ∧ 𝑘 ∈ (◡𝐴 “ (𝑆 ∖ {0}))) → 𝑘 ≤ sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, <
)) |
218 | 202, 203,
204, 216, 217 | syl31anc 1329 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, <
)) |
219 | 218, 53 | syl6breqr 4695 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀) |
220 | 219 | adantlr 751 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀) |
221 | 220 | adantlr 751 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀) |
222 | | simpllr 799 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑀 ≤ 𝑘) |
223 | 113 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ∈ ℝ) |
224 | 116 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑀 ∈ ℝ) |
225 | 223, 224 | letri3d 10179 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑘 = 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘))) |
226 | 221, 222,
225 | mpbir2and 957 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 = 𝑀) |
227 | 226 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑘 − 𝑀) = (𝑀 − 𝑀)) |
228 | 122 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑀 ∈ ℂ) |
229 | 228 | subidd 10380 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑀 − 𝑀) = 0) |
230 | 227, 229 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑘 − 𝑀) = 0) |
231 | 230 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑛↑(𝑘 − 𝑀)) = (𝑛↑0)) |
232 | 184 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑛 ∈ ℂ) |
233 | 232 | exp0d 13002 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑛↑0) = 1) |
234 | 231, 233 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → (𝑛↑(𝑘 − 𝑀)) = 1) |
235 | 234 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))) = ((𝐴‘𝑘) · 1)) |
236 | 226, 179 | sylibr 224 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ∈ {𝑀}) |
237 | 236 | iftrued 4094 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) = (𝐴‘𝑘)) |
238 | 199, 235,
237 | 3eqtr4d 2666 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) ∧ (𝐴‘𝑘) ≠ 0) → ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))) = if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
239 | 196, 238 | pm2.61dane 2881 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) ∧ 𝑛 ∈ ℕ) → ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))) = if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
240 | 239 | mpteq2dva 4744 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) = (𝑛 ∈ ℕ ↦ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0))) |
241 | | fconstmpt 5163 |
. . . . . . . . 9
⊢ (ℕ
× {if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)}) = (𝑛 ∈ ℕ ↦ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
242 | 240, 241 | syl6eqr 2674 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) = (ℕ × {if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)})) |
243 | | ifcl 4130 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑘) ∈ ℂ ∧ 0 ∈ ℂ)
→ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) ∈ ℂ) |
244 | 197, 7, 243 | sylancl 694 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) → if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) ∈ ℂ) |
245 | | 1z 11407 |
. . . . . . . . 9
⊢ 1 ∈
ℤ |
246 | 1 | eqimss2i 3660 |
. . . . . . . . . 10
⊢
(ℤ≥‘1) ⊆ ℕ |
247 | 246, 27 | climconst2 14279 |
. . . . . . . . 9
⊢
((if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) ∈ ℂ ∧ 1 ∈ ℤ)
→ (ℕ × {if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)}) ⇝ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
248 | 244, 245,
247 | sylancl 694 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) → (ℕ × {if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)}) ⇝ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
249 | 242, 248 | eqbrtrd 4675 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑀 ≤ 𝑘) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ⇝ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
250 | 183, 249,
113, 116 | ltlecasei 10145 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀)))) ⇝ if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
251 | | snex 4908 |
. . . . . . . 8
⊢ {0}
∈ V |
252 | 27, 251 | xpex 6962 |
. . . . . . 7
⊢ (ℕ
× {0}) ∈ V |
253 | 252 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (ℕ × {0})
∈ V) |
254 | 164 | anasss 679 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑚 ∈ ℕ)) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) ∈ ℂ) |
255 | | plyeq0.5 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0𝑝 =
(𝑧 ∈ ℂ ↦
Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
256 | 255 | fveq1d 6193 |
. . . . . . . . . . 11
⊢ (𝜑 →
(0𝑝‘𝑚) = ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑚)) |
257 | 256 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(0𝑝‘𝑚) = ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑚)) |
258 | 132 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
259 | | 0pval 23438 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℂ →
(0𝑝‘𝑚) = 0) |
260 | 258, 259 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) →
(0𝑝‘𝑚) = 0) |
261 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑚 → (𝑧↑𝑘) = (𝑚↑𝑘)) |
262 | 261 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑚 → ((𝐴‘𝑘) · (𝑧↑𝑘)) = ((𝐴‘𝑘) · (𝑚↑𝑘))) |
263 | 262 | sumeq2sdv 14435 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑚 → Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘))) |
264 | | eqid 2622 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℂ ↦
Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))) |
265 | | sumex 14418 |
. . . . . . . . . . . 12
⊢
Σ𝑘 ∈
(0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘)) ∈ V |
266 | 263, 264,
265 | fvmpt 6282 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℂ → ((𝑧 ∈ ℂ ↦
Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑚) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘))) |
267 | 258, 266 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))‘𝑚) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘))) |
268 | 257, 260,
267 | 3eqtr3d 2664 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘))) |
269 | 268 | oveq1d 6665 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0 / (𝑚↑𝑀)) = (Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀))) |
270 | | expcl 12878 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ℂ ∧ 𝑀 ∈ ℕ0)
→ (𝑚↑𝑀) ∈
ℂ) |
271 | 132, 86, 270 | syl2anr 495 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚↑𝑀) ∈ ℂ) |
272 | 139 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
273 | 258, 272,
159 | expne0d 13014 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚↑𝑀) ≠ 0) |
274 | 271, 273 | div0d 10800 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0 / (𝑚↑𝑀)) = 0) |
275 | | fzfid 12772 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0...𝑁) ∈ Fin) |
276 | | expcl 12878 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝑚↑𝑘) ∈
ℂ) |
277 | 258, 19, 276 | syl2an 494 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑚↑𝑘) ∈ ℂ) |
278 | 156, 277 | mulcld 10060 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑚↑𝑘)) ∈ ℂ) |
279 | 275, 271,
278, 273 | fsumdivc 14518 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀)) = Σ𝑘 ∈ (0...𝑁)(((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀))) |
280 | 269, 274,
279 | 3eqtr3d 2664 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 = Σ𝑘 ∈ (0...𝑁)(((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀))) |
281 | | fvconst2g 6467 |
. . . . . . . 8
⊢ ((0
∈ ℂ ∧ 𝑚
∈ ℕ) → ((ℕ × {0})‘𝑚) = 0) |
282 | 8, 281 | sylan 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ℕ ×
{0})‘𝑚) =
0) |
283 | 159 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑀 ∈ ℤ) |
284 | 48 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ) |
285 | 157, 158,
283, 284 | expsubd 13019 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑚↑(𝑘 − 𝑀)) = ((𝑚↑𝑘) / (𝑚↑𝑀))) |
286 | 285 | oveq2d 6666 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑚↑(𝑘 − 𝑀))) = ((𝐴‘𝑘) · ((𝑚↑𝑘) / (𝑚↑𝑀)))) |
287 | 271 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑚↑𝑀) ∈ ℂ) |
288 | 273 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑚↑𝑀) ≠ 0) |
289 | 156, 277,
287, 288 | divassd 10836 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → (((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀)) = ((𝐴‘𝑘) · ((𝑚↑𝑘) / (𝑚↑𝑀)))) |
290 | 286, 154,
289 | 3eqtr4d 2666 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = (((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀))) |
291 | 290 | sumeq2dv 14433 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...𝑁)((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚) = Σ𝑘 ∈ (0...𝑁)(((𝐴‘𝑘) · (𝑚↑𝑘)) / (𝑚↑𝑀))) |
292 | 280, 282,
291 | 3eqtr4d 2666 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ℕ ×
{0})‘𝑚) =
Σ𝑘 ∈ (0...𝑁)((𝑛 ∈ ℕ ↦ ((𝐴‘𝑘) · (𝑛↑(𝑘 − 𝑀))))‘𝑚)) |
293 | 1, 2, 3, 250, 253, 254, 292 | climfsum 14552 |
. . . . 5
⊢ (𝜑 → (ℕ × {0})
⇝ Σ𝑘 ∈
(0...𝑁)if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
294 | | suprleub 10989 |
. . . . . . . . . . . 12
⊢ ((((◡𝐴 “ (𝑆 ∖ {0})) ⊆ ℝ ∧ (◡𝐴 “ (𝑆 ∖ {0})) ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑥) ∧ 𝑁 ∈ ℝ) → (sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) ≤ 𝑁 ↔ ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑁)) |
295 | 201, 56, 82, 58, 294 | syl31anc 1329 |
. . . . . . . . . . 11
⊢ (𝜑 → (sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) ≤ 𝑁 ↔ ∀𝑧 ∈ (◡𝐴 “ (𝑆 ∖ {0}))𝑧 ≤ 𝑁)) |
296 | 78, 295 | mpbird 247 |
. . . . . . . . . 10
⊢ (𝜑 → sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) ≤ 𝑁) |
297 | 53, 296 | syl5eqbr 4688 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
298 | | nn0uz 11722 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
299 | 86, 298 | syl6eleq 2711 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘0)) |
300 | 57 | nn0zd 11480 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℤ) |
301 | | elfz5 12334 |
. . . . . . . . . 10
⊢ ((𝑀 ∈
(ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ≤ 𝑁)) |
302 | 299, 300,
301 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ≤ 𝑁)) |
303 | 297, 302 | mpbird 247 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ (0...𝑁)) |
304 | 303 | snssd 4340 |
. . . . . . 7
⊢ (𝜑 → {𝑀} ⊆ (0...𝑁)) |
305 | 18, 86 | ffvelrnd 6360 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴‘𝑀) ∈ ℂ) |
306 | | elsni 4194 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) |
307 | 306 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑘 ∈ {𝑀} → (𝐴‘𝑘) = (𝐴‘𝑀)) |
308 | 307 | eleq1d 2686 |
. . . . . . . . 9
⊢ (𝑘 ∈ {𝑀} → ((𝐴‘𝑘) ∈ ℂ ↔ (𝐴‘𝑀) ∈ ℂ)) |
309 | 305, 308 | syl5ibrcom 237 |
. . . . . . . 8
⊢ (𝜑 → (𝑘 ∈ {𝑀} → (𝐴‘𝑘) ∈ ℂ)) |
310 | 309 | ralrimiv 2965 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ {𝑀} (𝐴‘𝑘) ∈ ℂ) |
311 | 3 | olcd 408 |
. . . . . . 7
⊢ (𝜑 → ((0...𝑁) ⊆ (ℤ≥‘0)
∨ (0...𝑁) ∈
Fin)) |
312 | | sumss2 14457 |
. . . . . . 7
⊢ ((({𝑀} ⊆ (0...𝑁) ∧ ∀𝑘 ∈ {𝑀} (𝐴‘𝑘) ∈ ℂ) ∧ ((0...𝑁) ⊆
(ℤ≥‘0) ∨ (0...𝑁) ∈ Fin)) → Σ𝑘 ∈ {𝑀} (𝐴‘𝑘) = Σ𝑘 ∈ (0...𝑁)if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
313 | 304, 310,
311, 312 | syl21anc 1325 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ {𝑀} (𝐴‘𝑘) = Σ𝑘 ∈ (0...𝑁)if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0)) |
314 | | ltso 10118 |
. . . . . . . . 9
⊢ < Or
ℝ |
315 | 314 | supex 8369 |
. . . . . . . 8
⊢
sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) ∈
V |
316 | 53, 315 | eqeltri 2697 |
. . . . . . 7
⊢ 𝑀 ∈ V |
317 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → (𝐴‘𝑘) = (𝐴‘𝑀)) |
318 | 317 | sumsn 14475 |
. . . . . . 7
⊢ ((𝑀 ∈ V ∧ (𝐴‘𝑀) ∈ ℂ) → Σ𝑘 ∈ {𝑀} (𝐴‘𝑘) = (𝐴‘𝑀)) |
319 | 316, 305,
318 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ {𝑀} (𝐴‘𝑘) = (𝐴‘𝑀)) |
320 | 313, 319 | eqtr3d 2658 |
. . . . 5
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)if(𝑘 ∈ {𝑀}, (𝐴‘𝑘), 0) = (𝐴‘𝑀)) |
321 | 293, 320 | breqtrd 4679 |
. . . 4
⊢ (𝜑 → (ℕ × {0})
⇝ (𝐴‘𝑀)) |
322 | 246, 27 | climconst2 14279 |
. . . . 5
⊢ ((0
∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {0}) ⇝
0) |
323 | 7, 245, 322 | mp2an 708 |
. . . 4
⊢ (ℕ
× {0}) ⇝ 0 |
324 | | climuni 14283 |
. . . 4
⊢
(((ℕ × {0}) ⇝ (𝐴‘𝑀) ∧ (ℕ × {0}) ⇝ 0)
→ (𝐴‘𝑀) = 0) |
325 | 321, 323,
324 | sylancl 694 |
. . 3
⊢ (𝜑 → (𝐴‘𝑀) = 0) |
326 | | fvex 6201 |
. . . 4
⊢ (𝐴‘𝑀) ∈ V |
327 | 326 | elsn 4192 |
. . 3
⊢ ((𝐴‘𝑀) ∈ {0} ↔ (𝐴‘𝑀) = 0) |
328 | 325, 327 | sylibr 224 |
. 2
⊢ (𝜑 → (𝐴‘𝑀) ∈ {0}) |
329 | | elpreima 6337 |
. . . . . 6
⊢ (𝐴 Fn ℕ0 →
(𝑀 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑀 ∈ ℕ0 ∧ (𝐴‘𝑀) ∈ (𝑆 ∖ {0})))) |
330 | 66, 329 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝑀 ∈ (◡𝐴 “ (𝑆 ∖ {0})) ↔ (𝑀 ∈ ℕ0 ∧ (𝐴‘𝑀) ∈ (𝑆 ∖ {0})))) |
331 | 85, 330 | mpbid 222 |
. . . 4
⊢ (𝜑 → (𝑀 ∈ ℕ0 ∧ (𝐴‘𝑀) ∈ (𝑆 ∖ {0}))) |
332 | 331 | simprd 479 |
. . 3
⊢ (𝜑 → (𝐴‘𝑀) ∈ (𝑆 ∖ {0})) |
333 | 332 | eldifbd 3587 |
. 2
⊢ (𝜑 → ¬ (𝐴‘𝑀) ∈ {0}) |
334 | 328, 333 | pm2.65i 185 |
1
⊢ ¬
𝜑 |