| Step | Hyp | Ref
| Expression |
| 1 | | iprodgam.1 |
. . 3
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖
ℕ))) |
| 2 | | eflgam 27088 |
. . 3
⊢ (𝐴 ∈ (ℂ ∖
(ℤ ∖ ℕ)) → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴)) |
| 3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → (exp‘(log
Γ‘𝐴)) =
(Γ‘𝐴)) |
| 4 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑧 = 𝐴 → (𝑧 · (log‘((𝑘 + 1) / 𝑘))) = (𝐴 · (log‘((𝑘 + 1) / 𝑘)))) |
| 5 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑧 = 𝐴 → (𝑧 / 𝑘) = (𝐴 / 𝑘)) |
| 6 | 5 | fvoveq1d 7453 |
. . . . . . . . 9
⊢ (𝑧 = 𝐴 → (log‘((𝑧 / 𝑘) + 1)) = (log‘((𝐴 / 𝑘) + 1))) |
| 7 | 4, 6 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝑧 = 𝐴 → ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) |
| 8 | 7 | sumeq2sdv 15739 |
. . . . . . 7
⊢ (𝑧 = 𝐴 → Σ𝑘 ∈ ℕ ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) = Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) |
| 9 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑧 = 𝐴 → (log‘𝑧) = (log‘𝐴)) |
| 10 | 8, 9 | oveq12d 7449 |
. . . . . 6
⊢ (𝑧 = 𝐴 → (Σ𝑘 ∈ ℕ ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) − (log‘𝑧)) = (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) |
| 11 | | df-lgam 27062 |
. . . . . 6
⊢ log
Γ = (𝑧 ∈
(ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑘 ∈ ℕ ((𝑧 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝑧 / 𝑘) + 1))) − (log‘𝑧))) |
| 12 | | ovex 7464 |
. . . . . 6
⊢
(Σ𝑘 ∈
ℕ ((𝐴 ·
(log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴)) ∈ V |
| 13 | 10, 11, 12 | fvmpt 7016 |
. . . . 5
⊢ (𝐴 ∈ (ℂ ∖
(ℤ ∖ ℕ)) → (log Γ‘𝐴) = (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) |
| 14 | 1, 13 | syl 17 |
. . . 4
⊢ (𝜑 → (log Γ‘𝐴) = (Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) |
| 15 | 14 | fveq2d 6910 |
. . 3
⊢ (𝜑 → (exp‘(log
Γ‘𝐴)) =
(exp‘(Σ𝑘 ∈
ℕ ((𝐴 ·
(log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴)))) |
| 16 | | nnuz 12921 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
| 17 | | 1zzd 12648 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℤ) |
| 18 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1)) |
| 19 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → 𝑗 = 𝑘) |
| 20 | 18, 19 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((𝑗 + 1) / 𝑗) = ((𝑘 + 1) / 𝑘)) |
| 21 | 20 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (log‘((𝑗 + 1) / 𝑗)) = (log‘((𝑘 + 1) / 𝑘))) |
| 22 | 21 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → (𝐴 · (log‘((𝑗 + 1) / 𝑗))) = (𝐴 · (log‘((𝑘 + 1) / 𝑘)))) |
| 23 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝐴 / 𝑗) = (𝐴 / 𝑘)) |
| 24 | 23 | fvoveq1d 7453 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → (log‘((𝐴 / 𝑗) + 1)) = (log‘((𝐴 / 𝑘) + 1))) |
| 25 | 22, 24 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) |
| 26 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1)))) = (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1)))) |
| 27 | | ovex 7464 |
. . . . . . . 8
⊢ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ V |
| 28 | 25, 26, 27 | fvmpt 7016 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ → ((𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))‘𝑘) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) |
| 29 | 28 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))‘𝑘) = ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) |
| 30 | 1 | eldifad 3963 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 31 | 30 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ) |
| 32 | | peano2nn 12278 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
| 33 | 32 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) |
| 34 | 33 | nncnd 12282 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℂ) |
| 35 | | nncn 12274 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
| 36 | 35 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ) |
| 37 | | nnne0 12300 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
| 38 | 37 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0) |
| 39 | 34, 36, 38 | divcld 12043 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) ∈ ℂ) |
| 40 | 33 | nnne0d 12316 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ≠ 0) |
| 41 | 34, 36, 40, 38 | divne0d 12059 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) ≠ 0) |
| 42 | 39, 41 | logcld 26612 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘((𝑘 + 1) / 𝑘)) ∈ ℂ) |
| 43 | 31, 42 | mulcld 11281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (log‘((𝑘 + 1) / 𝑘))) ∈ ℂ) |
| 44 | 31, 36, 38 | divcld 12043 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℂ) |
| 45 | | 1cnd 11256 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈
ℂ) |
| 46 | 44, 45 | addcld 11280 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) ∈ ℂ) |
| 47 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖
ℕ))) |
| 48 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
| 49 | 47, 48 | dmgmdivn0 27071 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) ≠ 0) |
| 50 | 46, 49 | logcld 26612 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘((𝐴 / 𝑘) + 1)) ∈ ℂ) |
| 51 | 43, 50 | subcld 11620 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ ℂ) |
| 52 | 26, 1 | lgamcvg 27097 |
. . . . . . 7
⊢ (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ⇝ ((log Γ‘𝐴) + (log‘𝐴))) |
| 53 | | seqex 14044 |
. . . . . . . 8
⊢ seq1( + ,
(𝑗 ∈ ℕ ↦
((𝐴 ·
(log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ∈ V |
| 54 | | ovex 7464 |
. . . . . . . 8
⊢ ((log
Γ‘𝐴) +
(log‘𝐴)) ∈
V |
| 55 | 53, 54 | breldm 5919 |
. . . . . . 7
⊢ (seq1( +
, (𝑗 ∈ ℕ ↦
((𝐴 ·
(log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ⇝ ((log Γ‘𝐴) + (log‘𝐴)) → seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ∈ dom ⇝
) |
| 56 | 52, 55 | syl 17 |
. . . . . 6
⊢ (𝜑 → seq1( + , (𝑗 ∈ ℕ ↦ ((𝐴 · (log‘((𝑗 + 1) / 𝑗))) − (log‘((𝐴 / 𝑗) + 1))))) ∈ dom ⇝
) |
| 57 | 16, 17, 29, 51, 56 | isumcl 15797 |
. . . . 5
⊢ (𝜑 → Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ ℂ) |
| 58 | 1 | dmgmn0 27069 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≠ 0) |
| 59 | 30, 58 | logcld 26612 |
. . . . 5
⊢ (𝜑 → (log‘𝐴) ∈
ℂ) |
| 60 | | efsub 16136 |
. . . . 5
⊢
((Σ𝑘 ∈
ℕ ((𝐴 ·
(log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) ∈ ℂ ∧
(log‘𝐴) ∈
ℂ) → (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) = ((exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) / (exp‘(log‘𝐴)))) |
| 61 | 57, 59, 60 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) = ((exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) / (exp‘(log‘𝐴)))) |
| 62 | 16, 17, 29, 51, 56 | iprodefisum 35741 |
. . . . . 6
⊢ (𝜑 → ∏𝑘 ∈ ℕ (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = (exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))))) |
| 63 | | efsub 16136 |
. . . . . . . . 9
⊢ (((𝐴 · (log‘((𝑘 + 1) / 𝑘))) ∈ ℂ ∧ (log‘((𝐴 / 𝑘) + 1)) ∈ ℂ) →
(exp‘((𝐴 ·
(log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ((exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) / (exp‘(log‘((𝐴 / 𝑘) + 1))))) |
| 64 | 43, 50, 63 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ((exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) / (exp‘(log‘((𝐴 / 𝑘) + 1))))) |
| 65 | 36, 45, 36, 38 | divdird 12081 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) = ((𝑘 / 𝑘) + (1 / 𝑘))) |
| 66 | 36, 38 | dividd 12041 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 / 𝑘) = 1) |
| 67 | 66 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 / 𝑘) + (1 / 𝑘)) = (1 + (1 / 𝑘))) |
| 68 | 65, 67 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) / 𝑘) = (1 + (1 / 𝑘))) |
| 69 | 68 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘((𝑘 + 1) / 𝑘)) = (log‘(1 + (1 / 𝑘)))) |
| 70 | 69 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (log‘((𝑘 + 1) / 𝑘))) = (𝐴 · (log‘(1 + (1 / 𝑘))))) |
| 71 | 70 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) = (exp‘(𝐴 · (log‘(1 + (1 / 𝑘)))))) |
| 72 | | 1rp 13038 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℝ+ |
| 73 | 72 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈
ℝ+) |
| 74 | 48 | nnrpd 13075 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+) |
| 75 | 74 | rpreccld 13087 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈
ℝ+) |
| 76 | 73, 75 | rpaddcld 13092 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ∈
ℝ+) |
| 77 | 76 | rpcnd 13079 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ∈
ℂ) |
| 78 | 76 | rpne0d 13082 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 + (1 / 𝑘)) ≠ 0) |
| 79 | 77, 78, 31 | cxpefd 26754 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 + (1 / 𝑘))↑𝑐𝐴) = (exp‘(𝐴 · (log‘(1 + (1 / 𝑘)))))) |
| 80 | 71, 79 | eqtr4d 2780 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) = ((1 + (1 / 𝑘))↑𝑐𝐴)) |
| 81 | | eflog 26618 |
. . . . . . . . . . 11
⊢ ((((𝐴 / 𝑘) + 1) ∈ ℂ ∧ ((𝐴 / 𝑘) + 1) ≠ 0) →
(exp‘(log‘((𝐴 /
𝑘) + 1))) = ((𝐴 / 𝑘) + 1)) |
| 82 | 46, 49, 81 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) →
(exp‘(log‘((𝐴 /
𝑘) + 1))) = ((𝐴 / 𝑘) + 1)) |
| 83 | 44, 45 | addcomd 11463 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 / 𝑘) + 1) = (1 + (𝐴 / 𝑘))) |
| 84 | 82, 83 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) →
(exp‘(log‘((𝐴 /
𝑘) + 1))) = (1 + (𝐴 / 𝑘))) |
| 85 | 80, 84 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · (log‘((𝑘 + 1) / 𝑘)))) / (exp‘(log‘((𝐴 / 𝑘) + 1)))) = (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘)))) |
| 86 | 64, 85 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘)))) |
| 87 | 86 | prodeq2dv 15958 |
. . . . . 6
⊢ (𝜑 → ∏𝑘 ∈ ℕ (exp‘((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘)))) |
| 88 | 62, 87 | eqtr3d 2779 |
. . . . 5
⊢ (𝜑 → (exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘)))) |
| 89 | | eflog 26618 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) →
(exp‘(log‘𝐴)) =
𝐴) |
| 90 | 30, 58, 89 | syl2anc 584 |
. . . . 5
⊢ (𝜑 →
(exp‘(log‘𝐴)) =
𝐴) |
| 91 | 88, 90 | oveq12d 7449 |
. . . 4
⊢ (𝜑 → ((exp‘Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1)))) / (exp‘(log‘𝐴))) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) |
| 92 | 61, 91 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (exp‘(Σ𝑘 ∈ ℕ ((𝐴 · (log‘((𝑘 + 1) / 𝑘))) − (log‘((𝐴 / 𝑘) + 1))) − (log‘𝐴))) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) |
| 93 | 15, 92 | eqtrd 2777 |
. 2
⊢ (𝜑 → (exp‘(log
Γ‘𝐴)) =
(∏𝑘 ∈ ℕ
(((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) |
| 94 | 3, 93 | eqtr3d 2779 |
1
⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) |