Step | Hyp | Ref
| Expression |
1 | | xrltso 12875 |
. . . 4
⊢ < Or
ℝ* |
2 | 1 | a1i 11 |
. . 3
⊢ (𝜑 → < Or
ℝ*) |
3 | | nnuz 12621 |
. . . . 5
⊢ ℕ =
(ℤ≥‘1) |
4 | | 1zzd 12351 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℤ) |
5 | | esumpcvgval.1 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) |
6 | | esumpcvgval.2 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑙 → 𝐴 = 𝐵) |
7 | | eqcom 2745 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘) |
8 | | eqcom 2745 |
. . . . . . . . . . . 12
⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) |
9 | 6, 7, 8 | 3imtr3i 291 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝑘 → 𝐵 = 𝐴) |
10 | 9 | cbvmptv 5187 |
. . . . . . . . . 10
⊢ (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴) |
11 | 5, 10 | fmptd 6988 |
. . . . . . . . 9
⊢ (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞)) |
12 | 11 | ffvelrnda 6961 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞)) |
13 | | elrege0 13186 |
. . . . . . . . 9
⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥))) |
14 | 13 | simplbi 498 |
. . . . . . . 8
⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ) |
15 | 12, 14 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ) |
16 | 3, 4, 15 | serfre 13752 |
. . . . . 6
⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ) |
17 | 11 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞)) |
18 | | simpr 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
19 | 18 | peano2nnd 11990 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ) |
20 | 17, 19 | ffvelrnd 6962 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈
(0[,)+∞)) |
21 | | elrege0 13186 |
. . . . . . . . . 10
⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))) |
22 | 21 | simprbi 497 |
. . . . . . . . 9
⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → 0 ≤
((𝑙 ∈ ℕ ↦
𝐵)‘(𝑛 + 1))) |
23 | 20, 22 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))) |
24 | 16 | ffvelrnda 6961 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ) |
25 | 21 | simplbi 498 |
. . . . . . . . . 10
⊢ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ) |
26 | 20, 25 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ) |
27 | 24, 26 | addge01d 11563 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))) |
28 | 23, 27 | mpbid 231 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))) |
29 | 18, 3 | eleqtrdi 2849 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
(ℤ≥‘1)) |
30 | | seqp1 13736 |
. . . . . . . 8
⊢ (𝑛 ∈
(ℤ≥‘1) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))) |
31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))) |
32 | 28, 31 | breqtrrd 5102 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1))) |
33 | | simpr 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
34 | 10 | fvmpt2 6886 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) →
((𝑙 ∈ ℕ ↦
𝐵)‘𝑘) = 𝐴) |
35 | 33, 5, 34 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴) |
36 | | rge0ssre 13188 |
. . . . . . . . 9
⊢
(0[,)+∞) ⊆ ℝ |
37 | 36, 5 | sselid 3919 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ) |
38 | 16 | feqmptd 6837 |
. . . . . . . . . 10
⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))) |
39 | | simpll 764 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑) |
40 | | elfznn 13285 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ) |
41 | 40 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ) |
42 | 39, 41, 35 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴) |
43 | 37 | recnd 11003 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ) |
44 | 39, 41, 43 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ) |
45 | 42, 29, 44 | fsumser 15442 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
46 | 45 | eqcomd 2744 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴) |
47 | 46 | mpteq2dva 5174 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴)) |
48 | 38, 47 | eqtr2d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) = seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) |
49 | | esumpcvgval.3 |
. . . . . . . . 9
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ ) |
50 | 48, 49 | eqeltrrd 2840 |
. . . . . . . 8
⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝
) |
51 | 3, 4, 35, 37, 50 | isumrecl 15477 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ) |
52 | | 1zzd 12351 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈
ℤ) |
53 | | fzfid 13693 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin) |
54 | | fzssuz 13297 |
. . . . . . . . . . . 12
⊢
(1...𝑛) ⊆
(ℤ≥‘1) |
55 | 54, 3 | sseqtrri 3958 |
. . . . . . . . . . 11
⊢
(1...𝑛) ⊆
ℕ |
56 | 55 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆
ℕ) |
57 | 35 | adantlr 712 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴) |
58 | 37 | adantlr 712 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ) |
59 | 5 | adantlr 712 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) |
60 | | elrege0 13186 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ (0[,)+∞) ↔
(𝐴 ∈ ℝ ∧ 0
≤ 𝐴)) |
61 | 60 | simprbi 497 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ (0[,)+∞) → 0
≤ 𝐴) |
62 | 59, 61 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴) |
63 | 50 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝
) |
64 | 3, 52, 53, 56, 57, 58, 62, 63 | isumless 15557 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴) |
65 | 45, 64 | eqbrtrrd 5098 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴) |
66 | 65 | ralrimiva 3103 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴) |
67 | | brralrspcev 5134 |
. . . . . . 7
⊢
((Σ𝑘 ∈
ℕ 𝐴 ∈ ℝ
∧ ∀𝑛 ∈
ℕ (seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴) → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠) |
68 | 51, 66, 67 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠) |
69 | 3, 4, 16, 32, 68 | climsup 15381 |
. . . . 5
⊢ (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + ,
(𝑙 ∈ ℕ ↦
𝐵)), ℝ, <
)) |
70 | 3, 4, 69, 24 | climrecl 15292 |
. . . 4
⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈
ℝ) |
71 | 70 | rexrd 11025 |
. . 3
⊢ (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈
ℝ*) |
72 | | eqid 2738 |
. . . . . . 7
⊢ (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴) |
73 | | sumex 15399 |
. . . . . . 7
⊢
Σ𝑘 ∈
𝑏 𝐴 ∈ V |
74 | 72, 73 | elrnmpti 5869 |
. . . . . 6
⊢ (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) |
75 | | ssnnssfz 31108 |
. . . . . . . . . 10
⊢ (𝑏 ∈ (𝒫 ℕ ∩
Fin) → ∃𝑚 ∈
ℕ 𝑏 ⊆
(1...𝑚)) |
76 | | fzfid 13693 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin) |
77 | | elfznn 13285 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ) |
78 | 77, 5 | sylan2 593 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞)) |
79 | 60 | simplbi 498 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ (0[,)+∞) →
𝐴 ∈
ℝ) |
80 | 78, 79 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ) |
81 | 80 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ) |
82 | 78, 61 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴) |
83 | 82 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴) |
84 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚)) |
85 | 76, 81, 83, 84 | fsumless 15508 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑏 ⊆ (1...𝑚)) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴) |
86 | 85 | ex 413 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) |
87 | 86 | reximdv 3202 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) |
88 | 87 | imp 407 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴) |
89 | 75, 88 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
∃𝑚 ∈ ℕ
Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴) |
90 | | breq1 5077 |
. . . . . . . . . 10
⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) |
91 | 90 | rexbidv 3226 |
. . . . . . . . 9
⊢ (𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘 ∈ 𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) |
92 | 89, 91 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
(𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) |
93 | 92 | rexlimdva 3213 |
. . . . . . 7
⊢ (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) |
94 | 93 | imp 407 |
. . . . . 6
⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴) |
95 | 74, 94 | sylan2b 594 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴) |
96 | | simpr 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 = Σ𝑘 ∈ 𝑏 𝐴) |
97 | | inss2 4163 |
. . . . . . . . . . . . 13
⊢
(𝒫 ℕ ∩ Fin) ⊆ Fin |
98 | | simpr 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
𝑏 ∈ (𝒫 ℕ
∩ Fin)) |
99 | 97, 98 | sselid 3919 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
𝑏 ∈
Fin) |
100 | | simpll 764 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝜑) |
101 | | inss1 4162 |
. . . . . . . . . . . . . . . . 17
⊢
(𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ |
102 | | simplr 766 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝑏 ∈ (𝒫 ℕ ∩
Fin)) |
103 | 101, 102 | sselid 3919 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝑏 ∈ 𝒫 ℕ) |
104 | 103 | elpwid 4544 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝑏 ⊆ ℕ) |
105 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝑘 ∈ 𝑏) |
106 | 104, 105 | sseldd 3922 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝑘 ∈ ℕ) |
107 | 100, 106,
5 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝐴 ∈ (0[,)+∞)) |
108 | 107, 79 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝐴 ∈ ℝ) |
109 | 99, 108 | fsumrecl 15446 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ) |
110 | 109 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → Σ𝑘 ∈ 𝑏 𝐴 ∈ ℝ) |
111 | 96, 110 | eqeltrd 2839 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ) |
112 | 111 | r19.29an 3217 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘 ∈ 𝑏 𝐴) → 𝑥 ∈ ℝ) |
113 | 74, 112 | sylan2b 594 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ∈ ℝ) |
114 | 113 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ) |
115 | | fzfid 13693 |
. . . . . . . 8
⊢ (𝜑 → (1...𝑚) ∈ Fin) |
116 | 115, 80 | fsumrecl 15446 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ) |
117 | 116 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ) |
118 | 70 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈
ℝ) |
119 | | simprr 770 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴) |
120 | 16 | frnd 6608 |
. . . . . . . 8
⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆
ℝ) |
121 | 120 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ) |
122 | | 1nn 11984 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
123 | 122 | ne0ii 4271 |
. . . . . . . . 9
⊢ ℕ
≠ ∅ |
124 | | dm0rn0 5834 |
. . . . . . . . . . 11
⊢ (dom
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) = ∅
↔ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))
= ∅) |
125 | 16 | fdmd 6611 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ) |
126 | 125 | eqeq1d 2740 |
. . . . . . . . . . 11
⊢ (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ =
∅)) |
127 | 124, 126 | bitr3id 285 |
. . . . . . . . . 10
⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ =
∅)) |
128 | 127 | necon3bid 2988 |
. . . . . . . . 9
⊢ (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ
≠ ∅)) |
129 | 123, 128 | mpbiri 257 |
. . . . . . . 8
⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠
∅) |
130 | 129 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅) |
131 | | 1z 12350 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℤ |
132 | | seqfn 13733 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
ℤ → seq1( + , (𝑙
∈ ℕ ↦ 𝐵))
Fn (ℤ≥‘1)) |
133 | 131, 132 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ seq1( + ,
(𝑙 ∈ ℕ ↦
𝐵)) Fn
(ℤ≥‘1) |
134 | 3 | fneq2i 6531 |
. . . . . . . . . . . . . . 15
⊢ (seq1( +
, (𝑙 ∈ ℕ ↦
𝐵)) Fn ℕ ↔ seq1(
+ , (𝑙 ∈ ℕ
↦ 𝐵)) Fn
(ℤ≥‘1)) |
135 | 133, 134 | mpbir 230 |
. . . . . . . . . . . . . 14
⊢ seq1( + ,
(𝑙 ∈ ℕ ↦
𝐵)) Fn
ℕ |
136 | | dffn5 6828 |
. . . . . . . . . . . . . 14
⊢ (seq1( +
, (𝑙 ∈ ℕ ↦
𝐵)) Fn ℕ ↔ seq1(
+ , (𝑙 ∈ ℕ
↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( +
, (𝑙 ∈ ℕ ↦
𝐵))‘𝑛))) |
137 | 135, 136 | mpbi 229 |
. . . . . . . . . . . . 13
⊢ seq1( + ,
(𝑙 ∈ ℕ ↦
𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
138 | | fvex 6787 |
. . . . . . . . . . . . 13
⊢ (seq1( +
, (𝑙 ∈ ℕ ↦
𝐵))‘𝑛) ∈ V |
139 | 137, 138 | elrnmpti 5869 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
140 | | r19.29 3184 |
. . . . . . . . . . . . 13
⊢
((∀𝑛 ∈
ℕ (seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))) |
141 | | breq1 5077 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧 ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)) |
142 | 141 | biimparc 480 |
. . . . . . . . . . . . . 14
⊢ (((seq1(
+ , (𝑙 ∈ ℕ
↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠) |
143 | 142 | rexlimivw 3211 |
. . . . . . . . . . . . 13
⊢
(∃𝑛 ∈
ℕ ((seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠) |
144 | 140, 143 | syl 17 |
. . . . . . . . . . . 12
⊢
((∀𝑛 ∈
ℕ (seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧 ≤ 𝑠) |
145 | 139, 144 | sylan2b 594 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ 𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧 ≤ 𝑠) |
146 | 145 | ralrimiva 3103 |
. . . . . . . . . 10
⊢
(∀𝑛 ∈
ℕ (seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) |
147 | 146 | reximi 3178 |
. . . . . . . . 9
⊢
(∃𝑠 ∈
ℝ ∀𝑛 ∈
ℕ (seq1( + , (𝑙
∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) |
148 | 68, 147 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) |
149 | 148 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) |
150 | | simpr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ) |
151 | | simpll 764 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑) |
152 | 77 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ) |
153 | 151, 152,
35 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴) |
154 | 150, 3 | eleqtrdi 2849 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈
(ℤ≥‘1)) |
155 | 151, 152,
5 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞)) |
156 | 155, 79 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ) |
157 | 156 | recnd 11003 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ) |
158 | 153, 154,
157 | fsumser 15442 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) |
159 | | fveq2 6774 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) |
160 | 159 | rspceeqv 3575 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ℕ ∧
Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
161 | 150, 158,
160 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
162 | 137, 138 | elrnmpti 5869 |
. . . . . . . . 9
⊢
(Σ𝑘 ∈
(1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
163 | 161, 162 | sylibr 233 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) |
164 | 163 | ad2ant2r 744 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) |
165 | | suprub 11936 |
. . . . . . 7
⊢ (((ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) ⊆ ℝ
∧ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))
≠ ∅ ∧ ∃𝑠
∈ ℝ ∀𝑧
∈ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) |
166 | 121, 130,
149, 164, 165 | syl31anc 1372 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) |
167 | 114, 117,
118, 119, 166 | letrd 11132 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) |
168 | 95, 167 | rexlimddv 3220 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) |
169 | 70 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈
ℝ) |
170 | 113, 169 | lenltd 11121 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) → (𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ sup(ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)), ℝ, <
) < 𝑥)) |
171 | 168, 170 | mpbid 231 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥) |
172 | | simpr1r 1230 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0
≤ 𝑥 ∧ 𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, <
)) |
173 | 172 | 3anassrs 1359 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, <
)) |
174 | 71 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 = +∞) → sup(ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)), ℝ, <
) ∈ ℝ*) |
175 | | pnfnlt 12864 |
. . . . . . . 8
⊢ (sup(ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)), ℝ, <
) ∈ ℝ* → ¬ +∞ < sup(ran seq1( + ,
(𝑙 ∈ ℕ ↦
𝐵)), ℝ, <
)) |
176 | 174, 175 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 = +∞) → ¬
+∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) |
177 | | breq1 5077 |
. . . . . . . . 9
⊢ (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔
+∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) |
178 | 177 | notbid 318 |
. . . . . . . 8
⊢ (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬
+∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) |
179 | 178 | adantl 482 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 = +∞) → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬
+∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) |
180 | 176, 179 | mpbird 256 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 = +∞) → ¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, <
)) |
181 | 173, 180 | pm2.21dd 194 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 = +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦) |
182 | | simplll 772 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑) |
183 | | simpr1l 1229 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0
≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈
ℝ*) |
184 | 183 | 3anassrs 1359 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈
ℝ*) |
185 | | simplr 766 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤
𝑥) |
186 | | simpr 485 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞) |
187 | | 0xr 11022 |
. . . . . . . 8
⊢ 0 ∈
ℝ* |
188 | | pnfxr 11029 |
. . . . . . . 8
⊢ +∞
∈ ℝ* |
189 | | elico1 13122 |
. . . . . . . 8
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ*) →
(𝑥 ∈ (0[,)+∞)
↔ (𝑥 ∈
ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))) |
190 | 187, 188,
189 | mp2an 689 |
. . . . . . 7
⊢ (𝑥 ∈ (0[,)+∞) ↔
(𝑥 ∈
ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) |
191 | 184, 185,
186, 190 | syl3anbrc 1342 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈
(0[,)+∞)) |
192 | | simpr1r 1230 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0
≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, <
)) |
193 | 192 | 3anassrs 1359 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, <
)) |
194 | 120 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) ⊆
ℝ) |
195 | 129 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) ≠
∅) |
196 | 148 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
∃𝑠 ∈ ℝ
∀𝑧 ∈ ran seq1(
+ , (𝑙 ∈ ℕ
↦ 𝐵))𝑧 ≤ 𝑠) |
197 | 194, 195,
196 | 3jca 1127 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
(ran seq1( + , (𝑙 ∈
ℕ ↦ 𝐵)) ⊆
ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠)) |
198 | | simprl 768 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
𝑥 ∈
(0[,)+∞)) |
199 | 36, 198 | sselid 3919 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
𝑥 ∈
ℝ) |
200 | | simprr 770 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
𝑥 < sup(ran seq1( + ,
(𝑙 ∈ ℕ ↦
𝐵)), ℝ, <
)) |
201 | | suprlub 11939 |
. . . . . . . . 9
⊢ (((ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) ⊆ ℝ
∧ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))
≠ ∅ ∧ ∃𝑠
∈ ℝ ∀𝑧
∈ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ 𝑥 ∈ ℝ) → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)) |
202 | 201 | biimpa 477 |
. . . . . . . 8
⊢ ((((ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) ⊆ ℝ
∧ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))
≠ ∅ ∧ ∃𝑠
∈ ℝ ∀𝑧
∈ ran seq1( + , (𝑙
∈ ℕ ↦ 𝐵))𝑧 ≤ 𝑠) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) |
203 | 197, 199,
200, 202 | syl21anc 835 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
∃𝑦 ∈ ran seq1( +
, (𝑙 ∈ ℕ ↦
𝐵))𝑥 < 𝑦) |
204 | 40 | ssriv 3925 |
. . . . . . . . . . . . . . . . 17
⊢
(1...𝑛) ⊆
ℕ |
205 | | ovex 7308 |
. . . . . . . . . . . . . . . . . 18
⊢
(1...𝑛) ∈
V |
206 | 205 | elpw 4537 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑛) ∈
𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ) |
207 | 204, 206 | mpbir 230 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑛) ∈
𝒫 ℕ |
208 | | fzfi 13692 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑛) ∈
Fin |
209 | | elin 3903 |
. . . . . . . . . . . . . . . 16
⊢
((1...𝑛) ∈
(𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin)) |
210 | 207, 208,
209 | mpbir2an 708 |
. . . . . . . . . . . . . . 15
⊢
(1...𝑛) ∈
(𝒫 ℕ ∩ Fin) |
211 | 210 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩
Fin)) |
212 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
213 | 45 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
214 | 212, 213 | eqtr4d 2781 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) |
215 | | sumeq1 15400 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (1...𝑛) → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴) |
216 | 215 | rspceeqv 3575 |
. . . . . . . . . . . . . 14
⊢
(((1...𝑛) ∈
(𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴) |
217 | 211, 214,
216 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴) |
218 | 217 | ex 413 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)) |
219 | 218 | rexlimdva 3213 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴)) |
220 | 137, 138 | elrnmpti 5869 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) |
221 | 72, 73 | elrnmpti 5869 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘 ∈ 𝑏 𝐴) |
222 | 219, 220,
221 | 3imtr4g 296 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴))) |
223 | 222 | ssrdv 3927 |
. . . . . . . . 9
⊢ (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴)) |
224 | | ssrexv 3988 |
. . . . . . . . 9
⊢ (ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)) ⊆ ran
(𝑏 ∈ (𝒫
ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)) |
225 | 223, 224 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦)) |
226 | 225 | imp 407 |
. . . . . . 7
⊢ ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦) |
227 | 203, 226 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴)𝑥 < 𝑦) |
228 | 182, 191,
193, 227 | syl12anc 834 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) ∧ 𝑥 < +∞) →
∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴)𝑥 < 𝑦) |
229 | | simplrl 774 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) → 𝑥 ∈
ℝ*) |
230 | | xrlelttric 31075 |
. . . . . . . 8
⊢
((+∞ ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (+∞
≤ 𝑥 ∨ 𝑥 <
+∞)) |
231 | 188, 230 | mpan 687 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ*
→ (+∞ ≤ 𝑥
∨ 𝑥 <
+∞)) |
232 | | xgepnf 12899 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ*
→ (+∞ ≤ 𝑥
↔ 𝑥 =
+∞)) |
233 | 232 | orbi1d 914 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ*
→ ((+∞ ≤ 𝑥
∨ 𝑥 < +∞)
↔ (𝑥 = +∞ ∨
𝑥 <
+∞))) |
234 | 231, 233 | mpbid 231 |
. . . . . 6
⊢ (𝑥 ∈ ℝ*
→ (𝑥 = +∞ ∨
𝑥 <
+∞)) |
235 | 229, 234 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 <
+∞)) |
236 | 181, 228,
235 | mpjaodan 956 |
. . . 4
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0
≤ 𝑥) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦) |
237 | | 0elpw 5278 |
. . . . . . . . 9
⊢ ∅
∈ 𝒫 ℕ |
238 | | 0fin 8954 |
. . . . . . . . 9
⊢ ∅
∈ Fin |
239 | | elin 3903 |
. . . . . . . . 9
⊢ (∅
∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ
∧ ∅ ∈ Fin)) |
240 | 237, 238,
239 | mpbir2an 708 |
. . . . . . . 8
⊢ ∅
∈ (𝒫 ℕ ∩ Fin) |
241 | | sum0 15433 |
. . . . . . . . 9
⊢
Σ𝑘 ∈
∅ 𝐴 =
0 |
242 | 241 | eqcomi 2747 |
. . . . . . . 8
⊢ 0 =
Σ𝑘 ∈ ∅
𝐴 |
243 | | sumeq1 15400 |
. . . . . . . . 9
⊢ (𝑏 = ∅ → Σ𝑘 ∈ 𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴) |
244 | 243 | rspceeqv 3575 |
. . . . . . . 8
⊢ ((∅
∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 =
Σ𝑘 ∈ 𝑏 𝐴) |
245 | 240, 242,
244 | mp2an 689 |
. . . . . . 7
⊢
∃𝑏 ∈
(𝒫 ℕ ∩ Fin)0 = Σ𝑘 ∈ 𝑏 𝐴 |
246 | 72, 73 | elrnmpti 5869 |
. . . . . . 7
⊢ (0 ∈
ran (𝑏 ∈ (𝒫
ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 =
Σ𝑘 ∈ 𝑏 𝐴) |
247 | 245, 246 | mpbir 230 |
. . . . . 6
⊢ 0 ∈
ran (𝑏 ∈ (𝒫
ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) |
248 | | breq2 5078 |
. . . . . . 7
⊢ (𝑦 = 0 → (𝑥 < 𝑦 ↔ 𝑥 < 0)) |
249 | 248 | rspcev 3561 |
. . . . . 6
⊢ ((0
∈ ran (𝑏 ∈
(𝒫 ℕ ∩ Fin) ↦ Σ𝑘 ∈ 𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦) |
250 | 247, 249 | mpan 687 |
. . . . 5
⊢ (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴)𝑥 < 𝑦) |
251 | 250 | adantl 482 |
. . . 4
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧
𝑥 < 0) →
∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴)𝑥 < 𝑦) |
252 | | xrlelttric 31075 |
. . . . . 6
⊢ ((0
∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (0 ≤
𝑥 ∨ 𝑥 < 0)) |
253 | 187, 252 | mpan 687 |
. . . . 5
⊢ (𝑥 ∈ ℝ*
→ (0 ≤ 𝑥 ∨ 𝑥 < 0)) |
254 | 253 | ad2antrl 725 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0
≤ 𝑥 ∨ 𝑥 < 0)) |
255 | 236, 251,
254 | mpjaodan 956 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) →
∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴)𝑥 < 𝑦) |
256 | 2, 71, 171, 255 | eqsupd 9216 |
. 2
⊢ (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩
Fin) ↦ Σ𝑘
∈ 𝑏 𝐴), ℝ*, < ) = sup(ran
seq1( + , (𝑙 ∈ ℕ
↦ 𝐵)), ℝ, <
)) |
257 | | nfv 1917 |
. . 3
⊢
Ⅎ𝑘𝜑 |
258 | | nfcv 2907 |
. . 3
⊢
Ⅎ𝑘ℕ |
259 | | nnex 11979 |
. . . 4
⊢ ℕ
∈ V |
260 | 259 | a1i 11 |
. . 3
⊢ (𝜑 → ℕ ∈
V) |
261 | | icossicc 13168 |
. . . 4
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
262 | 261, 5 | sselid 3919 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
263 | | elex 3450 |
. . . . . 6
⊢ (𝑏 ∈ (𝒫 ℕ ∩
Fin) → 𝑏 ∈
V) |
264 | 263 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
𝑏 ∈
V) |
265 | 107 | fmpttd 6989 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
(𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞)) |
266 | | esumpfinvallem 32042 |
. . . . 5
⊢ ((𝑏 ∈ V ∧ (𝑘 ∈ 𝑏 ↦ 𝐴):𝑏⟶(0[,)+∞)) →
(ℂfld Σg (𝑘 ∈ 𝑏 ↦ 𝐴)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑏 ↦ 𝐴))) |
267 | 264, 265,
266 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
(ℂfld Σg (𝑘 ∈ 𝑏 ↦ 𝐴)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑏 ↦ 𝐴))) |
268 | 108 | recnd 11003 |
. . . . 5
⊢ (((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧
𝑘 ∈ 𝑏) → 𝐴 ∈ ℂ) |
269 | 99, 268 | gsumfsum 20665 |
. . . 4
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
(ℂfld Σg (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴) |
270 | 267, 269 | eqtr3d 2780 |
. . 3
⊢ ((𝜑 ∧ 𝑏 ∈ (𝒫 ℕ ∩ Fin)) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ 𝑏 ↦ 𝐴)) = Σ𝑘 ∈ 𝑏 𝐴) |
271 | 257, 258,
260, 262, 270 | esumval 32014 |
. 2
⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦
Σ𝑘 ∈ 𝑏 𝐴), ℝ*, <
)) |
272 | 3, 4, 35, 43, 69 | isumclim 15469 |
. 2
⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) |
273 | 256, 271,
272 | 3eqtr4d 2788 |
1
⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |