Proof of Theorem cxp2limlem
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 0red 11264 | . 2
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 0 ∈
ℝ) | 
| 2 |  | 2rp 13039 | . . . . 5
⊢ 2 ∈
ℝ+ | 
| 3 |  | rplogcl 26646 | . . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (log‘𝐴) ∈
ℝ+) | 
| 4 |  | 2z 12649 | . . . . . 6
⊢ 2 ∈
ℤ | 
| 5 |  | rpexpcl 14121 | . . . . . 6
⊢
(((log‘𝐴)
∈ ℝ+ ∧ 2 ∈ ℤ) → ((log‘𝐴)↑2) ∈
ℝ+) | 
| 6 | 3, 4, 5 | sylancl 586 | . . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → ((log‘𝐴)↑2) ∈
ℝ+) | 
| 7 |  | rpdivcl 13060 | . . . . 5
⊢ ((2
∈ ℝ+ ∧ ((log‘𝐴)↑2) ∈ ℝ+) →
(2 / ((log‘𝐴)↑2)) ∈
ℝ+) | 
| 8 | 2, 6, 7 | sylancr 587 | . . . 4
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (2 /
((log‘𝐴)↑2))
∈ ℝ+) | 
| 9 | 8 | rpcnd 13079 | . . 3
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (2 /
((log‘𝐴)↑2))
∈ ℂ) | 
| 10 |  | divrcnv 15888 | . . 3
⊢ ((2 /
((log‘𝐴)↑2))
∈ ℂ → (𝑛
∈ ℝ+ ↦ ((2 / ((log‘𝐴)↑2)) / 𝑛)) ⇝𝑟
0) | 
| 11 | 9, 10 | syl 17 | . 2
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (𝑛 ∈ ℝ+
↦ ((2 / ((log‘𝐴)↑2)) / 𝑛)) ⇝𝑟
0) | 
| 12 | 8 | rpred 13077 | . . 3
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (2 /
((log‘𝐴)↑2))
∈ ℝ) | 
| 13 |  | rerpdivcl 13065 | . . 3
⊢ (((2 /
((log‘𝐴)↑2))
∈ ℝ ∧ 𝑛
∈ ℝ+) → ((2 / ((log‘𝐴)↑2)) / 𝑛) ∈ ℝ) | 
| 14 | 12, 13 | sylan 580 | . 2
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((2 /
((log‘𝐴)↑2)) /
𝑛) ∈
ℝ) | 
| 15 |  | simpr 484 | . . . 4
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝑛 ∈
ℝ+) | 
| 16 |  | simpl 482 | . . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 𝐴 ∈ ℝ) | 
| 17 |  | 1red 11262 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 1 ∈
ℝ) | 
| 18 |  | 0lt1 11785 | . . . . . . . 8
⊢ 0 <
1 | 
| 19 | 18 | a1i 11 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 0 <
1) | 
| 20 |  | simpr 484 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 1 < 𝐴) | 
| 21 | 1, 17, 16, 19, 20 | lttrd 11422 | . . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 0 < 𝐴) | 
| 22 | 16, 21 | elrpd 13074 | . . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 𝐴 ∈
ℝ+) | 
| 23 |  | rpre 13043 | . . . . 5
⊢ (𝑛 ∈ ℝ+
→ 𝑛 ∈
ℝ) | 
| 24 |  | rpcxpcl 26718 | . . . . 5
⊢ ((𝐴 ∈ ℝ+
∧ 𝑛 ∈ ℝ)
→ (𝐴↑𝑐𝑛) ∈
ℝ+) | 
| 25 | 22, 23, 24 | syl2an 596 | . . . 4
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝐴↑𝑐𝑛) ∈
ℝ+) | 
| 26 | 15, 25 | rpdivcld 13094 | . . 3
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / (𝐴↑𝑐𝑛)) ∈
ℝ+) | 
| 27 | 26 | rpred 13077 | . 2
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / (𝐴↑𝑐𝑛)) ∈ ℝ) | 
| 28 | 3 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) →
(log‘𝐴) ∈
ℝ+) | 
| 29 | 15, 28 | rpmulcld 13093 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 · (log‘𝐴)) ∈
ℝ+) | 
| 30 | 29 | rpred 13077 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 · (log‘𝐴)) ∈
ℝ) | 
| 31 | 30 | resqcld 14165 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛 · (log‘𝐴))↑2) ∈
ℝ) | 
| 32 | 31 | rehalfcld 12513 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (((𝑛 · (log‘𝐴))↑2) / 2) ∈
ℝ) | 
| 33 |  | 1rp 13038 | . . . . . . . . . . 11
⊢ 1 ∈
ℝ+ | 
| 34 |  | rpaddcl 13057 | . . . . . . . . . . 11
⊢ ((1
∈ ℝ+ ∧ (𝑛 · (log‘𝐴)) ∈ ℝ+) → (1 +
(𝑛 ·
(log‘𝐴))) ∈
ℝ+) | 
| 35 | 33, 29, 34 | sylancr 587 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (1 +
(𝑛 ·
(log‘𝐴))) ∈
ℝ+) | 
| 36 | 35 | rpred 13077 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (1 +
(𝑛 ·
(log‘𝐴))) ∈
ℝ) | 
| 37 | 36, 32 | readdcld 11290 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((1 +
(𝑛 ·
(log‘𝐴))) + (((𝑛 · (log‘𝐴))↑2) / 2)) ∈
ℝ) | 
| 38 | 30 | reefcld 16124 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) →
(exp‘(𝑛 ·
(log‘𝐴))) ∈
ℝ) | 
| 39 | 32, 35 | ltaddrp2d 13111 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (((𝑛 · (log‘𝐴))↑2) / 2) < ((1 +
(𝑛 ·
(log‘𝐴))) + (((𝑛 · (log‘𝐴))↑2) /
2))) | 
| 40 |  | efgt1p2 16150 | . . . . . . . . 9
⊢ ((𝑛 · (log‘𝐴)) ∈ ℝ+
→ ((1 + (𝑛 ·
(log‘𝐴))) + (((𝑛 · (log‘𝐴))↑2) / 2)) <
(exp‘(𝑛 ·
(log‘𝐴)))) | 
| 41 | 29, 40 | syl 17 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((1 +
(𝑛 ·
(log‘𝐴))) + (((𝑛 · (log‘𝐴))↑2) / 2)) <
(exp‘(𝑛 ·
(log‘𝐴)))) | 
| 42 | 32, 37, 38, 39, 41 | lttrd 11422 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (((𝑛 · (log‘𝐴))↑2) / 2) <
(exp‘(𝑛 ·
(log‘𝐴)))) | 
| 43 | 23 | adantl 481 | . . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝑛 ∈
ℝ) | 
| 44 | 43 | recnd 11289 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝑛 ∈
ℂ) | 
| 45 | 44 | sqcld 14184 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛↑2) ∈
ℂ) | 
| 46 |  | 2cnd 12344 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 2 ∈
ℂ) | 
| 47 | 6 | adantr 480 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) →
((log‘𝐴)↑2)
∈ ℝ+) | 
| 48 | 47 | rpcnd 13079 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) →
((log‘𝐴)↑2)
∈ ℂ) | 
| 49 |  | 2ne0 12370 | . . . . . . . . . 10
⊢ 2 ≠
0 | 
| 50 | 49 | a1i 11 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 2 ≠
0) | 
| 51 | 47 | rpne0d 13082 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) →
((log‘𝐴)↑2) ≠
0) | 
| 52 | 45, 46, 48, 50, 51 | divdiv2d 12075 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑2) / (2 /
((log‘𝐴)↑2))) =
(((𝑛↑2) ·
((log‘𝐴)↑2)) /
2)) | 
| 53 | 3 | rpcnd 13079 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (log‘𝐴) ∈
ℂ) | 
| 54 | 53 | adantr 480 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) →
(log‘𝐴) ∈
ℂ) | 
| 55 | 44, 54 | sqmuld 14198 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛 · (log‘𝐴))↑2) = ((𝑛↑2) ·
((log‘𝐴)↑2))) | 
| 56 | 55 | oveq1d 7446 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (((𝑛 · (log‘𝐴))↑2) / 2) = (((𝑛↑2) ·
((log‘𝐴)↑2)) /
2)) | 
| 57 | 52, 56 | eqtr4d 2780 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑2) / (2 /
((log‘𝐴)↑2))) =
(((𝑛 ·
(log‘𝐴))↑2) /
2)) | 
| 58 | 16 | recnd 11289 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → 𝐴 ∈ ℂ) | 
| 59 | 58 | adantr 480 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈
ℂ) | 
| 60 | 22 | adantr 480 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈
ℝ+) | 
| 61 | 60 | rpne0d 13082 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝐴 ≠ 0) | 
| 62 | 59, 61, 44 | cxpefd 26754 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝐴↑𝑐𝑛) = (exp‘(𝑛 · (log‘𝐴)))) | 
| 63 | 42, 57, 62 | 3brtr4d 5175 | . . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑2) / (2 /
((log‘𝐴)↑2)))
< (𝐴↑𝑐𝑛)) | 
| 64 |  | rpexpcl 14121 | . . . . . . . . 9
⊢ ((𝑛 ∈ ℝ+
∧ 2 ∈ ℤ) → (𝑛↑2) ∈
ℝ+) | 
| 65 | 15, 4, 64 | sylancl 586 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛↑2) ∈
ℝ+) | 
| 66 | 8 | adantr 480 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (2 /
((log‘𝐴)↑2))
∈ ℝ+) | 
| 67 | 65, 66 | rpdivcld 13094 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑2) / (2 /
((log‘𝐴)↑2)))
∈ ℝ+) | 
| 68 | 67, 25, 15 | ltdiv2d 13100 | . . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (((𝑛↑2) / (2 /
((log‘𝐴)↑2)))
< (𝐴↑𝑐𝑛) ↔ (𝑛 / (𝐴↑𝑐𝑛)) < (𝑛 / ((𝑛↑2) / (2 / ((log‘𝐴)↑2)))))) | 
| 69 | 63, 68 | mpbid 232 | . . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / (𝐴↑𝑐𝑛)) < (𝑛 / ((𝑛↑2) / (2 / ((log‘𝐴)↑2))))) | 
| 70 | 9 | adantr 480 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (2 /
((log‘𝐴)↑2))
∈ ℂ) | 
| 71 | 65 | rpne0d 13082 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛↑2) ≠
0) | 
| 72 | 66 | rpne0d 13082 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (2 /
((log‘𝐴)↑2))
≠ 0) | 
| 73 | 44, 45, 70, 71, 72 | divdiv2d 12075 | . . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / ((𝑛↑2) / (2 / ((log‘𝐴)↑2)))) = ((𝑛 · (2 / ((log‘𝐴)↑2))) / (𝑛↑2))) | 
| 74 | 44 | sqvald 14183 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛↑2) = (𝑛 · 𝑛)) | 
| 75 | 74 | oveq2d 7447 | . . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛 · (2 / ((log‘𝐴)↑2))) / (𝑛↑2)) = ((𝑛 · (2 / ((log‘𝐴)↑2))) / (𝑛 · 𝑛))) | 
| 76 |  | rpne0 13051 | . . . . . . . 8
⊢ (𝑛 ∈ ℝ+
→ 𝑛 ≠
0) | 
| 77 | 76 | adantl 481 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 𝑛 ≠ 0) | 
| 78 | 70, 44, 44, 77, 77 | divcan5d 12069 | . . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → ((𝑛 · (2 / ((log‘𝐴)↑2))) / (𝑛 · 𝑛)) = ((2 / ((log‘𝐴)↑2)) / 𝑛)) | 
| 79 | 73, 75, 78 | 3eqtrd 2781 | . . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / ((𝑛↑2) / (2 / ((log‘𝐴)↑2)))) = ((2 / ((log‘𝐴)↑2)) / 𝑛)) | 
| 80 | 69, 79 | breqtrd 5169 | . . . 4
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / (𝐴↑𝑐𝑛)) < ((2 / ((log‘𝐴)↑2)) / 𝑛)) | 
| 81 | 27, 14, 80 | ltled 11409 | . . 3
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → (𝑛 / (𝐴↑𝑐𝑛)) ≤ ((2 / ((log‘𝐴)↑2)) / 𝑛)) | 
| 82 | 81 | adantrr 717 | . 2
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ (𝑛 ∈ ℝ+
∧ 0 ≤ 𝑛)) →
(𝑛 / (𝐴↑𝑐𝑛)) ≤ ((2 / ((log‘𝐴)↑2)) / 𝑛)) | 
| 83 | 26 | rpge0d 13081 | . . 3
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ 𝑛 ∈ ℝ+) → 0 ≤
(𝑛 / (𝐴↑𝑐𝑛))) | 
| 84 | 83 | adantrr 717 | . 2
⊢ (((𝐴 ∈ ℝ ∧ 1 <
𝐴) ∧ (𝑛 ∈ ℝ+
∧ 0 ≤ 𝑛)) → 0
≤ (𝑛 / (𝐴↑𝑐𝑛))) | 
| 85 | 1, 1, 11, 14, 27, 82, 84 | rlimsqz2 15687 | 1
⊢ ((𝐴 ∈ ℝ ∧ 1 <
𝐴) → (𝑛 ∈ ℝ+
↦ (𝑛 / (𝐴↑𝑐𝑛))) ⇝𝑟
0) |