| Step | Hyp | Ref
| Expression |
| 1 | | nnuz 12921 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
| 2 | | 1zzd 12648 |
. . . 4
⊢ (⊤
→ 1 ∈ ℤ) |
| 3 | | ax-1cn 11213 |
. . . . 5
⊢ 1 ∈
ℂ |
| 4 | 1 | eqimss2i 4045 |
. . . . . 6
⊢
(ℤ≥‘1) ⊆ ℕ |
| 5 | | nnex 12272 |
. . . . . 6
⊢ ℕ
∈ V |
| 6 | 4, 5 | climconst2 15584 |
. . . . 5
⊢ ((1
∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {1}) ⇝
1) |
| 7 | 3, 2, 6 | sylancr 587 |
. . . 4
⊢ (⊤
→ (ℕ × {1}) ⇝ 1) |
| 8 | | ovexd 7466 |
. . . 4
⊢ (⊤
→ ((ℕ × {1}) ∘f + ((ℕ × {𝐴}) ∘f ·
𝐺)) ∈
V) |
| 9 | | basellem7.2 |
. . . . . . 7
⊢ 𝐴 ∈ ℂ |
| 10 | 4, 5 | climconst2 15584 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℤ) → (ℕ × {𝐴}) ⇝ 𝐴) |
| 11 | 9, 2, 10 | sylancr 587 |
. . . . . 6
⊢ (⊤
→ (ℕ × {𝐴}) ⇝ 𝐴) |
| 12 | | ovexd 7466 |
. . . . . 6
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) ∈ V) |
| 13 | | basel.g |
. . . . . . . 8
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 ·
𝑛) + 1))) |
| 14 | 13 | basellem6 27129 |
. . . . . . 7
⊢ 𝐺 ⇝ 0 |
| 15 | 14 | a1i 11 |
. . . . . 6
⊢ (⊤
→ 𝐺 ⇝
0) |
| 16 | 9 | elexi 3503 |
. . . . . . . . 9
⊢ 𝐴 ∈ V |
| 17 | 16 | fconst 6794 |
. . . . . . . 8
⊢ (ℕ
× {𝐴}):ℕ⟶{𝐴} |
| 18 | 9 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ 𝐴 ∈
ℂ) |
| 19 | 18 | snssd 4809 |
. . . . . . . 8
⊢ (⊤
→ {𝐴} ⊆
ℂ) |
| 20 | | fss 6752 |
. . . . . . . 8
⊢
(((ℕ × {𝐴}):ℕ⟶{𝐴} ∧ {𝐴} ⊆ ℂ) → (ℕ ×
{𝐴}):ℕ⟶ℂ) |
| 21 | 17, 19, 20 | sylancr 587 |
. . . . . . 7
⊢ (⊤
→ (ℕ × {𝐴}):ℕ⟶ℂ) |
| 22 | 21 | ffvelcdmda 7104 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {𝐴})‘𝑘) ∈ ℂ) |
| 23 | | 2nn 12339 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℕ |
| 24 | 23 | a1i 11 |
. . . . . . . . . . . 12
⊢ (⊤
→ 2 ∈ ℕ) |
| 25 | | nnmulcl 12290 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℕ ∧ 𝑛
∈ ℕ) → (2 · 𝑛) ∈ ℕ) |
| 26 | 24, 25 | sylan 580 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (2 · 𝑛) ∈ ℕ) |
| 27 | 26 | peano2nnd 12283 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑛
∈ ℕ) → ((2 · 𝑛) + 1) ∈ ℕ) |
| 28 | 27 | nnrecred 12317 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ) |
| 29 | 28 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑛
∈ ℕ) → (1 / ((2 · 𝑛) + 1)) ∈ ℂ) |
| 30 | 29, 13 | fmptd 7134 |
. . . . . . 7
⊢ (⊤
→ 𝐺:ℕ⟶ℂ) |
| 31 | 30 | ffvelcdmda 7104 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℂ) |
| 32 | 21 | ffnd 6737 |
. . . . . . 7
⊢ (⊤
→ (ℕ × {𝐴}) Fn ℕ) |
| 33 | 30 | ffnd 6737 |
. . . . . . 7
⊢ (⊤
→ 𝐺 Fn
ℕ) |
| 34 | 5 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ ℕ ∈ V) |
| 35 | | inidm 4227 |
. . . . . . 7
⊢ (ℕ
∩ ℕ) = ℕ |
| 36 | | eqidd 2738 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {𝐴})‘𝑘) = ((ℕ × {𝐴})‘𝑘)) |
| 37 | | eqidd 2738 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) = (𝐺‘𝑘)) |
| 38 | 32, 33, 34, 34, 35, 36, 37 | ofval 7708 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘) = (((ℕ × {𝐴})‘𝑘) · (𝐺‘𝑘))) |
| 39 | 1, 2, 11, 12, 15, 22, 31, 38 | climmul 15669 |
. . . . 5
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) ⇝ (𝐴 · 0)) |
| 40 | 9 | mul01i 11451 |
. . . . 5
⊢ (𝐴 · 0) =
0 |
| 41 | 39, 40 | breqtrdi 5184 |
. . . 4
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) ⇝ 0) |
| 42 | | 1ex 11257 |
. . . . . . 7
⊢ 1 ∈
V |
| 43 | 42 | fconst 6794 |
. . . . . 6
⊢ (ℕ
× {1}):ℕ⟶{1} |
| 44 | 3 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ ℂ) |
| 45 | 44 | snssd 4809 |
. . . . . 6
⊢ (⊤
→ {1} ⊆ ℂ) |
| 46 | | fss 6752 |
. . . . . 6
⊢
(((ℕ × {1}):ℕ⟶{1} ∧ {1} ⊆ ℂ)
→ (ℕ × {1}):ℕ⟶ℂ) |
| 47 | 43, 45, 46 | sylancr 587 |
. . . . 5
⊢ (⊤
→ (ℕ × {1}):ℕ⟶ℂ) |
| 48 | 47 | ffvelcdmda 7104 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {1})‘𝑘) ∈ ℂ) |
| 49 | | mulcl 11239 |
. . . . . . 7
⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
| 50 | 49 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℂ ∧ 𝑦
∈ ℂ)) → (𝑥
· 𝑦) ∈
ℂ) |
| 51 | 50, 21, 30, 34, 34, 35 | off 7715 |
. . . . 5
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺):ℕ⟶ℂ) |
| 52 | 51 | ffvelcdmda 7104 |
. . . 4
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘) ∈ ℂ) |
| 53 | 43 | a1i 11 |
. . . . . 6
⊢ (⊤
→ (ℕ × {1}):ℕ⟶{1}) |
| 54 | 53 | ffnd 6737 |
. . . . 5
⊢ (⊤
→ (ℕ × {1}) Fn ℕ) |
| 55 | 51 | ffnd 6737 |
. . . . 5
⊢ (⊤
→ ((ℕ × {𝐴}) ∘f · 𝐺) Fn ℕ) |
| 56 | | eqidd 2738 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((ℕ × {1})‘𝑘) = ((ℕ × {1})‘𝑘)) |
| 57 | | eqidd 2738 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘) = (((ℕ × {𝐴}) ∘f · 𝐺)‘𝑘)) |
| 58 | 54, 55, 34, 34, 35, 56, 57 | ofval 7708 |
. . . 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 15668 |
. . 3
⊢ (⊤
→ ((ℕ × {1}) ∘f + ((ℕ × {𝐴}) ∘f ·
𝐺)) ⇝ (1 +
0)) |
| 60 | 59 | mptru 1547 |
. 2
⊢ ((ℕ
× {1}) ∘f + ((ℕ × {𝐴}) ∘f · 𝐺)) ⇝ (1 +
0) |
| 61 | | 1p0e1 12390 |
. 2
⊢ (1 + 0) =
1 |
| 62 | 60, 61 | breqtri 5168 |
1
⊢ ((ℕ
× {1}) ∘f + ((ℕ × {𝐴}) ∘f · 𝐺)) ⇝ 1 |