| Step | Hyp | Ref
| Expression |
| 1 | | stirlinglem14.1 |
. . 3
⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)))) |
| 2 | | stirlinglem14.2 |
. . 3
⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛))) |
| 3 | 1, 2 | stirlinglem13 46101 |
. 2
⊢
∃𝑑 ∈
ℝ 𝐵 ⇝ 𝑑 |
| 4 | | simpl 482 |
. . . . 5
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → 𝑑 ∈ ℝ) |
| 5 | 4 | rpefcld 16141 |
. . . 4
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → (exp‘𝑑) ∈
ℝ+) |
| 6 | | nnuz 12921 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
| 7 | | 1zzd 12648 |
. . . . . 6
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → 1 ∈ ℤ) |
| 8 | | efcn 26487 |
. . . . . . 7
⊢ exp
∈ (ℂ–cn→ℂ) |
| 9 | 8 | a1i 11 |
. . . . . 6
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → exp ∈ (ℂ–cn→ℂ)) |
| 10 | | nnnn0 12533 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
| 11 | | faccl 14322 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ0
→ (!‘𝑛) ∈
ℕ) |
| 12 | | nncn 12274 |
. . . . . . . . . . . . 13
⊢
((!‘𝑛) ∈
ℕ → (!‘𝑛)
∈ ℂ) |
| 13 | 10, 11, 12 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ →
(!‘𝑛) ∈
ℂ) |
| 14 | | 2cnd 12344 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 2 ∈
ℂ) |
| 15 | | nncn 12274 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
| 16 | 14, 15 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → (2
· 𝑛) ∈
ℂ) |
| 17 | 16 | sqrtcld 15476 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ →
(√‘(2 · 𝑛)) ∈ ℂ) |
| 18 | | epr 16244 |
. . . . . . . . . . . . . . . . 17
⊢ e ∈
ℝ+ |
| 19 | | rpcn 13045 |
. . . . . . . . . . . . . . . . 17
⊢ (e ∈
ℝ+ → e ∈ ℂ) |
| 20 | 18, 19 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ e ∈
ℂ |
| 21 | 20 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → e ∈
ℂ) |
| 22 | | 0re 11263 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ∈
ℝ |
| 23 | | epos 16243 |
. . . . . . . . . . . . . . . . 17
⊢ 0 <
e |
| 24 | 22, 23 | gtneii 11373 |
. . . . . . . . . . . . . . . 16
⊢ e ≠
0 |
| 25 | 24 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → e ≠
0) |
| 26 | 15, 21, 25 | divcld 12043 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈
ℂ) |
| 27 | 26, 10 | expcld 14186 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℂ) |
| 28 | 17, 27 | mulcld 11281 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ →
((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℂ) |
| 29 | | 2rp 13039 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℝ+ |
| 30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → 2 ∈
ℝ+) |
| 31 | | nnrp 13046 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
| 32 | 30, 31 | rpmulcld 13093 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → (2
· 𝑛) ∈
ℝ+) |
| 33 | 32 | sqrtgt0d 15451 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → 0 <
(√‘(2 · 𝑛))) |
| 34 | 33 | gt0ne0d 11827 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ →
(√‘(2 · 𝑛)) ≠ 0) |
| 35 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
| 36 | 15, 21, 35, 25 | divne0d 12059 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → (𝑛 / e) ≠ 0) |
| 37 | | nnz 12634 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
| 38 | 26, 36, 37 | expne0d 14192 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ≠ 0) |
| 39 | 17, 27, 34, 38 | mulne0d 11915 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ →
((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)) ≠ 0) |
| 40 | 13, 28, 39 | divcld 12043 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ →
((!‘𝑛) /
((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))) ∈ ℂ) |
| 41 | 1 | fvmpt2 7027 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧
((!‘𝑛) /
((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))) ∈ ℂ) → (𝐴‘𝑛) = ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) |
| 42 | 40, 41 | mpdan 687 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (𝐴‘𝑛) = ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) |
| 43 | 42, 40 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ → (𝐴‘𝑛) ∈ ℂ) |
| 44 | | nnne0 12300 |
. . . . . . . . . . . 12
⊢
((!‘𝑛) ∈
ℕ → (!‘𝑛)
≠ 0) |
| 45 | 10, 11, 44 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ →
(!‘𝑛) ≠
0) |
| 46 | 13, 28, 45, 39 | divne0d 12059 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ →
((!‘𝑛) /
((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))) ≠ 0) |
| 47 | 42, 46 | eqnetrd 3008 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ → (𝐴‘𝑛) ≠ 0) |
| 48 | 43, 47 | logcld 26612 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ →
(log‘(𝐴‘𝑛)) ∈
ℂ) |
| 49 | 2, 48 | fmpti 7132 |
. . . . . . 7
⊢ 𝐵:ℕ⟶ℂ |
| 50 | 49 | a1i 11 |
. . . . . 6
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → 𝐵:ℕ⟶ℂ) |
| 51 | | simpr 484 |
. . . . . 6
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → 𝐵 ⇝ 𝑑) |
| 52 | 4 | recnd 11289 |
. . . . . 6
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → 𝑑 ∈ ℂ) |
| 53 | 6, 7, 9, 50, 51, 52 | climcncf 24926 |
. . . . 5
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → (exp ∘ 𝐵) ⇝ (exp‘𝑑)) |
| 54 | 8 | elexi 3503 |
. . . . . . . . 9
⊢ exp
∈ V |
| 55 | | nnex 12272 |
. . . . . . . . . . 11
⊢ ℕ
∈ V |
| 56 | 55 | mptex 7243 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦
(log‘(𝐴‘𝑛))) ∈ V |
| 57 | 2, 56 | eqeltri 2837 |
. . . . . . . . 9
⊢ 𝐵 ∈ V |
| 58 | 54, 57 | coex 7952 |
. . . . . . . 8
⊢ (exp
∘ 𝐵) ∈
V |
| 59 | 58 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ (exp ∘ 𝐵)
∈ V) |
| 60 | 55 | mptex 7243 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦
((!‘𝑛) /
((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) ∈ V |
| 61 | 1, 60 | eqeltri 2837 |
. . . . . . . 8
⊢ 𝐴 ∈ V |
| 62 | 61 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ 𝐴 ∈
V) |
| 63 | | 1zzd 12648 |
. . . . . . 7
⊢ (⊤
→ 1 ∈ ℤ) |
| 64 | 2 | funmpt2 6605 |
. . . . . . . . . 10
⊢ Fun 𝐵 |
| 65 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ) |
| 66 | | rabid2 3470 |
. . . . . . . . . . . . 13
⊢ (ℕ
= {𝑛 ∈ ℕ ∣
(log‘(𝐴‘𝑛)) ∈ V} ↔
∀𝑛 ∈ ℕ
(log‘(𝐴‘𝑛)) ∈ V) |
| 67 | 1 | stirlinglem2 46090 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → (𝐴‘𝑛) ∈
ℝ+) |
| 68 | | relogcl 26617 |
. . . . . . . . . . . . . 14
⊢ ((𝐴‘𝑛) ∈ ℝ+ →
(log‘(𝐴‘𝑛)) ∈
ℝ) |
| 69 | | elex 3501 |
. . . . . . . . . . . . . 14
⊢
((log‘(𝐴‘𝑛)) ∈ ℝ → (log‘(𝐴‘𝑛)) ∈ V) |
| 70 | 67, 68, 69 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ →
(log‘(𝐴‘𝑛)) ∈ V) |
| 71 | 66, 70 | mprgbir 3068 |
. . . . . . . . . . . 12
⊢ ℕ =
{𝑛 ∈ ℕ ∣
(log‘(𝐴‘𝑛)) ∈ V} |
| 72 | 2 | dmmpt 6260 |
. . . . . . . . . . . 12
⊢ dom 𝐵 = {𝑛 ∈ ℕ ∣ (log‘(𝐴‘𝑛)) ∈ V} |
| 73 | 71, 72 | eqtr4i 2768 |
. . . . . . . . . . 11
⊢ ℕ =
dom 𝐵 |
| 74 | 65, 73 | eleqtrdi 2851 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → 𝑘 ∈ dom 𝐵) |
| 75 | | fvco 7007 |
. . . . . . . . . 10
⊢ ((Fun
𝐵 ∧ 𝑘 ∈ dom 𝐵) → ((exp ∘ 𝐵)‘𝑘) = (exp‘(𝐵‘𝑘))) |
| 76 | 64, 74, 75 | sylancr 587 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ → ((exp
∘ 𝐵)‘𝑘) = (exp‘(𝐵‘𝑘))) |
| 77 | 1 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛))))) |
| 78 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → 𝑛 = 𝑘) |
| 79 | 78 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → (!‘𝑛) = (!‘𝑘)) |
| 80 | 78 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → (2 · 𝑛) = (2 · 𝑘)) |
| 81 | 80 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → (√‘(2 · 𝑛)) = (√‘(2 ·
𝑘))) |
| 82 | 78 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → (𝑛 / e) = (𝑘 / e)) |
| 83 | 82, 78 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → ((𝑛 / e)↑𝑛) = ((𝑘 / e)↑𝑘)) |
| 84 | 81, 83 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)) = ((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘))) |
| 85 | 79, 84 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧ 𝑛 = 𝑘) → ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))) = ((!‘𝑘) / ((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘)))) |
| 86 | | nnnn0 12533 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
| 87 | | faccl 14322 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ∈
ℕ) |
| 88 | | nncn 12274 |
. . . . . . . . . . . . . . . 16
⊢
((!‘𝑘) ∈
ℕ → (!‘𝑘)
∈ ℂ) |
| 89 | 86, 87, 88 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ →
(!‘𝑘) ∈
ℂ) |
| 90 | | 2cnd 12344 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 2 ∈
ℂ) |
| 91 | | nncn 12274 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
| 92 | 90, 91 | mulcld 11281 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (2
· 𝑘) ∈
ℂ) |
| 93 | 92 | sqrtcld 15476 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ →
(√‘(2 · 𝑘)) ∈ ℂ) |
| 94 | 20 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → e ∈
ℂ) |
| 95 | 24 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → e ≠
0) |
| 96 | 91, 94, 95 | divcld 12043 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (𝑘 / e) ∈
ℂ) |
| 97 | 96, 86 | expcld 14186 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → ((𝑘 / e)↑𝑘) ∈ ℂ) |
| 98 | 93, 97 | mulcld 11281 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ →
((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘)) ∈ ℂ) |
| 99 | 29 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ → 2 ∈
ℝ+) |
| 100 | | nnrp 13046 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ+) |
| 101 | 99, 100 | rpmulcld 13093 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → (2
· 𝑘) ∈
ℝ+) |
| 102 | 101 | sqrtgt0d 15451 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → 0 <
(√‘(2 · 𝑘))) |
| 103 | 102 | gt0ne0d 11827 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ →
(√‘(2 · 𝑘)) ≠ 0) |
| 104 | | nnne0 12300 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
| 105 | 91, 94, 104, 95 | divne0d 12059 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (𝑘 / e) ≠ 0) |
| 106 | | nnz 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℤ) |
| 107 | 96, 105, 106 | expne0d 14192 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ → ((𝑘 / e)↑𝑘) ≠ 0) |
| 108 | 93, 97, 103, 107 | mulne0d 11915 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ →
((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘)) ≠ 0) |
| 109 | 89, 98, 108 | divcld 12043 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ →
((!‘𝑘) /
((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘))) ∈ ℂ) |
| 110 | 77, 85, 65, 109 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) = ((!‘𝑘) / ((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘)))) |
| 111 | 110, 109 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ∈ ℂ) |
| 112 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢
((!‘𝑘) ∈
ℕ → (!‘𝑘)
≠ 0) |
| 113 | 86, 87, 112 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ →
(!‘𝑘) ≠
0) |
| 114 | 89, 98, 113, 108 | divne0d 12059 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ →
((!‘𝑘) /
((√‘(2 · 𝑘)) · ((𝑘 / e)↑𝑘))) ≠ 0) |
| 115 | 110, 114 | eqnetrd 3008 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) ≠ 0) |
| 116 | 111, 115 | logcld 26612 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ →
(log‘(𝐴‘𝑘)) ∈
ℂ) |
| 117 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛𝑘 |
| 118 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛log |
| 119 | | nfmpt1 5250 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛(𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 ·
𝑛)) · ((𝑛 / e)↑𝑛)))) |
| 120 | 1, 119 | nfcxfr 2903 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝐴 |
| 121 | 120, 117 | nffv 6916 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛(𝐴‘𝑘) |
| 122 | 118, 121 | nffv 6916 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛(log‘(𝐴‘𝑘)) |
| 123 | | 2fveq3 6911 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (log‘(𝐴‘𝑛)) = (log‘(𝐴‘𝑘))) |
| 124 | 117, 122,
123, 2 | fvmptf 7037 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
(log‘(𝐴‘𝑘)) ∈ ℂ) → (𝐵‘𝑘) = (log‘(𝐴‘𝑘))) |
| 125 | 116, 124 | mpdan 687 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → (𝐵‘𝑘) = (log‘(𝐴‘𝑘))) |
| 126 | 125 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ →
(exp‘(𝐵‘𝑘)) =
(exp‘(log‘(𝐴‘𝑘)))) |
| 127 | | eflog 26618 |
. . . . . . . . . 10
⊢ (((𝐴‘𝑘) ∈ ℂ ∧ (𝐴‘𝑘) ≠ 0) → (exp‘(log‘(𝐴‘𝑘))) = (𝐴‘𝑘)) |
| 128 | 111, 115,
127 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ →
(exp‘(log‘(𝐴‘𝑘))) = (𝐴‘𝑘)) |
| 129 | 76, 126, 128 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ → ((exp
∘ 𝐵)‘𝑘) = (𝐴‘𝑘)) |
| 130 | 129 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((exp ∘ 𝐵)‘𝑘) = (𝐴‘𝑘)) |
| 131 | 6, 59, 62, 63, 130 | climeq 15603 |
. . . . . 6
⊢ (⊤
→ ((exp ∘ 𝐵)
⇝ (exp‘𝑑)
↔ 𝐴 ⇝
(exp‘𝑑))) |
| 132 | 131 | mptru 1547 |
. . . . 5
⊢ ((exp
∘ 𝐵) ⇝
(exp‘𝑑) ↔ 𝐴 ⇝ (exp‘𝑑)) |
| 133 | 53, 132 | sylib 218 |
. . . 4
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → 𝐴 ⇝ (exp‘𝑑)) |
| 134 | | breq2 5147 |
. . . . 5
⊢ (𝑐 = (exp‘𝑑) → (𝐴 ⇝ 𝑐 ↔ 𝐴 ⇝ (exp‘𝑑))) |
| 135 | 134 | rspcev 3622 |
. . . 4
⊢
(((exp‘𝑑)
∈ ℝ+ ∧ 𝐴 ⇝ (exp‘𝑑)) → ∃𝑐 ∈ ℝ+ 𝐴 ⇝ 𝑐) |
| 136 | 5, 133, 135 | syl2anc 584 |
. . 3
⊢ ((𝑑 ∈ ℝ ∧ 𝐵 ⇝ 𝑑) → ∃𝑐 ∈ ℝ+ 𝐴 ⇝ 𝑐) |
| 137 | 136 | rexlimiva 3147 |
. 2
⊢
(∃𝑑 ∈
ℝ 𝐵 ⇝ 𝑑 → ∃𝑐 ∈ ℝ+
𝐴 ⇝ 𝑐) |
| 138 | 3, 137 | ax-mp 5 |
1
⊢
∃𝑐 ∈
ℝ+ 𝐴
⇝ 𝑐 |