Step | Hyp | Ref
| Expression |
1 | | nnuz 12364 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
2 | | 1zzd 12095 |
. . . 4
⊢ (⊤
→ 1 ∈ ℤ) |
3 | | ax-1cn 10674 |
. . . . 5
⊢ 1 ∈
ℂ |
4 | 1 | eqimss2i 3937 |
. . . . . 6
⊢
(ℤ≥‘1) ⊆ ℕ |
5 | | nnex 11723 |
. . . . . 6
⊢ ℕ
∈ V |
6 | 4, 5 | climconst2 14996 |
. . . . 5
⊢ ((1
∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {1}) ⇝
1) |
7 | 3, 2, 6 | sylancr 590 |
. . . 4
⊢ (⊤
→ (ℕ × {1}) ⇝ 1) |
8 | | ovexd 7206 |
. . . 4
⊢ (⊤
→ ((ℕ × {1}) ∘f + ((ℕ × {𝐴}) ∘f ·
𝐺)) ∈
V) |
9 | | basellem7.2 |
. . . . . . 7
⊢ 𝐴 ∈ ℂ |
10 | 4, 5 | climconst2 14996 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℤ) → (ℕ × {𝐴}) ⇝ 𝐴) |
11 | 9, 2, 10 | sylancr 590 |
. . . . . 6
⊢ (⊤
→ (ℕ × {𝐴}) ⇝ 𝐴) |
12 | | ovexd 7206 |
. . . . . 6
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) ∈ V) |
13 | | basel.g |
. . . . . . . 8
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 ·
𝑛) + 1))) |
14 | 13 | basellem6 25823 |
. . . . . . 7
⊢ 𝐺 ⇝ 0 |
15 | 14 | a1i 11 |
. . . . . 6
⊢ (⊤
→ 𝐺 ⇝
0) |
16 | 9 | elexi 3417 |
. . . . . . . . 9
⊢ 𝐴 ∈ V |
17 | 16 | fconst 6565 |
. . . . . . . 8
⊢ (ℕ
× {𝐴}):ℕ⟶{𝐴} |
18 | 9 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ 𝐴 ∈
ℂ) |
19 | 18 | snssd 4698 |
. . . . . . . 8
⊢ (⊤
→ {𝐴} ⊆
ℂ) |
20 | | fss 6522 |
. . . . . . . 8
⊢
(((ℕ × {𝐴}):ℕ⟶{𝐴} ∧ {𝐴} ⊆ ℂ) → (ℕ ×
{𝐴}):ℕ⟶ℂ) |
21 | 17, 19, 20 | sylancr 590 |
. . . . . . 7
⊢ (⊤
→ (ℕ × {𝐴}):ℕ⟶ℂ) |
22 | 21 | ffvelrnda 6862 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {𝐴})‘𝑘) ∈ ℂ) |
23 | | 2nn 11790 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℕ |
24 | 23 | a1i 11 |
. . . . . . . . . . . 12
⊢ (⊤
→ 2 ∈ ℕ) |
25 | | nnmulcl 11741 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℕ ∧ 𝑛
∈ ℕ) → (2 · 𝑛) ∈ ℕ) |
26 | 24, 25 | sylan 583 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (2 · 𝑛) ∈ ℕ) |
27 | 26 | peano2nnd 11734 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((2 · 𝑛) + 1) ∈ ℕ) |
28 | 27 | nnrecred 11768 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ) |
29 | 28 | recnd 10748 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / ((2 · 𝑛) + 1)) ∈ ℂ) |
30 | 29, 13 | fmptd 6889 |
. . . . . . 7
⊢ (⊤
→ 𝐺:ℕ⟶ℂ) |
31 | 30 | ffvelrnda 6862 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℂ) |
32 | 21 | ffnd 6506 |
. . . . . . 7
⊢ (⊤
→ (ℕ × {𝐴}) Fn ℕ) |
33 | 30 | ffnd 6506 |
. . . . . . 7
⊢ (⊤
→ 𝐺 Fn
ℕ) |
34 | 5 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ ℕ ∈ V) |
35 | | inidm 4110 |
. . . . . . 7
⊢ (ℕ
∩ ℕ) = ℕ |
36 | | eqidd 2739 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {𝐴})‘𝑘) = ((ℕ × {𝐴})‘𝑘)) |
37 | | eqidd 2739 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) = (𝐺‘𝑘)) |
38 | 32, 33, 34, 34, 35, 36, 37 | ofval 7436 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘) = (((ℕ × {𝐴})‘𝑘) · (𝐺‘𝑘))) |
39 | 1, 2, 11, 12, 15, 22, 31, 38 | climmul 15081 |
. . . . 5
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) ⇝ (𝐴 · 0)) |
40 | 9 | mul01i 10909 |
. . . . 5
⊢ (𝐴 · 0) =
0 |
41 | 39, 40 | breqtrdi 5072 |
. . . 4
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) ⇝ 0) |
42 | | 1ex 10716 |
. . . . . . 7
⊢ 1 ∈
V |
43 | 42 | fconst 6565 |
. . . . . 6
⊢ (ℕ
× {1}):ℕ⟶{1} |
44 | 3 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ ℂ) |
45 | 44 | snssd 4698 |
. . . . . 6
⊢ (⊤
→ {1} ⊆ ℂ) |
46 | | fss 6522 |
. . . . . 6
⊢
(((ℕ × {1}):ℕ⟶{1} ∧ {1} ⊆ ℂ)
→ (ℕ × {1}):ℕ⟶ℂ) |
47 | 43, 45, 46 | sylancr 590 |
. . . . 5
⊢ (⊤
→ (ℕ × {1}):ℕ⟶ℂ) |
48 | 47 | ffvelrnda 6862 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {1})‘𝑘) ∈ ℂ) |
49 | | mulcl 10700 |
. . . . . . 7
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
50 | 49 | adantl 485 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℂ ∧ 𝑦
∈ ℂ)) → (𝑥
· 𝑦) ∈
ℂ) |
51 | 50, 21, 30, 34, 34, 35 | off 7443 |
. . . . 5
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺):ℕ⟶ℂ) |
52 | 51 | ffvelrnda 6862 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘) ∈ ℂ) |
53 | 43 | a1i 11 |
. . . . . 6
⊢ (⊤
→ (ℕ × {1}):ℕ⟶{1}) |
54 | 53 | ffnd 6506 |
. . . . 5
⊢ (⊤
→ (ℕ × {1}) Fn ℕ) |
55 | 51 | ffnd 6506 |
. . . . 5
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) Fn ℕ) |
56 | | eqidd 2739 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {1})‘𝑘) = ((ℕ × {1})‘𝑘)) |
57 | | eqidd 2739 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘) = (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘)) |
58 | 54, 55, 34, 34, 35, 56, 57 | ofval 7436 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {1}) ∘f + ((ℕ
× {𝐴})
∘f · 𝐺))‘𝑘) = (((ℕ × {1})‘𝑘) + (((ℕ × {𝐴}) ∘f ·
𝐺)‘𝑘))) |
59 | 1, 2, 7, 8, 41, 48, 52, 58 | climadd 15080 |
. . 3
⊢ (⊤
→ ((ℕ × {1}) ∘f + ((ℕ × {𝐴}) ∘f ·
𝐺)) ⇝ (1 +
0)) |
60 | 59 | mptru 1549 |
. 2
⊢ ((ℕ
× {1}) ∘f + ((ℕ × {𝐴}) ∘f · 𝐺)) ⇝ (1 +
0) |
61 | | 1p0e1 11841 |
. 2
⊢ (1 + 0) =
1 |
62 | 60, 61 | breqtri 5056 |
1
⊢ ((ℕ
× {1}) ∘f + ((ℕ × {𝐴}) ∘f · 𝐺)) ⇝ 1 |