Proof of Theorem cxp2lim
| Step | Hyp | Ref
| Expression |
| 1 | | 1re 11261 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
| 2 | | elicopnf 13485 |
. . . . . . . 8
⊢ (1 ∈
ℝ → (𝑛 ∈
(1[,)+∞) ↔ (𝑛
∈ ℝ ∧ 1 ≤ 𝑛))) |
| 3 | 1, 2 | ax-mp 5 |
. . . . . . 7
⊢ (𝑛 ∈ (1[,)+∞) ↔
(𝑛 ∈ ℝ ∧ 1
≤ 𝑛)) |
| 4 | 3 | simplbi 497 |
. . . . . 6
⊢ (𝑛 ∈ (1[,)+∞) →
𝑛 ∈
ℝ) |
| 5 | | 0red 11264 |
. . . . . . 7
⊢ (𝑛 ∈ (1[,)+∞) → 0
∈ ℝ) |
| 6 | | 1red 11262 |
. . . . . . 7
⊢ (𝑛 ∈ (1[,)+∞) → 1
∈ ℝ) |
| 7 | | 0lt1 11785 |
. . . . . . . 8
⊢ 0 <
1 |
| 8 | 7 | a1i 11 |
. . . . . . 7
⊢ (𝑛 ∈ (1[,)+∞) → 0
< 1) |
| 9 | 3 | simprbi 496 |
. . . . . . 7
⊢ (𝑛 ∈ (1[,)+∞) → 1
≤ 𝑛) |
| 10 | 5, 6, 4, 8, 9 | ltletrd 11421 |
. . . . . 6
⊢ (𝑛 ∈ (1[,)+∞) → 0
< 𝑛) |
| 11 | 4, 10 | elrpd 13074 |
. . . . 5
⊢ (𝑛 ∈ (1[,)+∞) →
𝑛 ∈
ℝ+) |
| 12 | 11 | ssriv 3987 |
. . . 4
⊢
(1[,)+∞) ⊆ ℝ+ |
| 13 | | resmpt 6055 |
. . . 4
⊢
((1[,)+∞) ⊆ ℝ+ → ((𝑛 ∈ ℝ+ ↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ↾ (1[,)+∞)) = (𝑛 ∈ (1[,)+∞) ↦
((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)))) |
| 14 | 12, 13 | ax-mp 5 |
. . 3
⊢ ((𝑛 ∈ ℝ+
↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ↾ (1[,)+∞)) = (𝑛 ∈ (1[,)+∞) ↦
((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) |
| 15 | | 0red 11264 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 0 ∈
ℝ) |
| 16 | 12 | a1i 11 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (1[,)+∞)
⊆ ℝ+) |
| 17 | | rpre 13043 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℝ+
→ 𝑛 ∈
ℝ) |
| 18 | 17 | adantl 481 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 𝑛 ∈
ℝ) |
| 19 | | rpge0 13048 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℝ+
→ 0 ≤ 𝑛) |
| 20 | 19 | adantl 481 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 0 ≤
𝑛) |
| 21 | | simpl2 1193 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 𝐵 ∈
ℝ) |
| 22 | | 0red 11264 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 0 ∈
ℝ) |
| 23 | | 1red 11262 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 1 ∈
ℝ) |
| 24 | 7 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 0 <
1) |
| 25 | | simpl3 1194 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 1 <
𝐵) |
| 26 | 22, 23, 21, 24, 25 | lttrd 11422 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 0 <
𝐵) |
| 27 | 21, 26 | elrpd 13074 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 𝐵 ∈
ℝ+) |
| 28 | 27, 18 | rpcxpcld 26775 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝐵↑𝑐𝑛) ∈
ℝ+) |
| 29 | | simp1 1137 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 𝐴 ∈ ℝ) |
| 30 | | ifcl 4571 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 1 ∈
ℝ) → if(1 ≤ 𝐴, 𝐴, 1) ∈ ℝ) |
| 31 | 29, 1, 30 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → if(1 ≤ 𝐴, 𝐴, 1) ∈ ℝ) |
| 32 | | 1red 11262 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 1 ∈
ℝ) |
| 33 | 7 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 0 <
1) |
| 34 | | max1 13227 |
. . . . . . . . . . . . . . 15
⊢ ((1
∈ ℝ ∧ 𝐴
∈ ℝ) → 1 ≤ if(1 ≤ 𝐴, 𝐴, 1)) |
| 35 | 1, 29, 34 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 1 ≤ if(1 ≤
𝐴, 𝐴, 1)) |
| 36 | 15, 32, 31, 33, 35 | ltletrd 11421 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 0 < if(1 ≤
𝐴, 𝐴, 1)) |
| 37 | 31, 36 | elrpd 13074 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → if(1 ≤ 𝐴, 𝐴, 1) ∈
ℝ+) |
| 38 | 37 | rprecred 13088 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (1 / if(1 ≤
𝐴, 𝐴, 1)) ∈ ℝ) |
| 39 | 38 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (1 / if(1
≤ 𝐴, 𝐴, 1)) ∈ ℝ) |
| 40 | 28, 39 | rpcxpcld 26775 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1))) ∈
ℝ+) |
| 41 | 31 | recnd 11289 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → if(1 ≤ 𝐴, 𝐴, 1) ∈ ℂ) |
| 42 | 41 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → if(1 ≤
𝐴, 𝐴, 1) ∈ ℂ) |
| 43 | 18, 20, 40, 42 | divcxpd 26764 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1))))↑𝑐if(1 ≤
𝐴, 𝐴, 1)) = ((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐if(1 ≤
𝐴, 𝐴, 1)))) |
| 44 | 37 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → if(1 ≤
𝐴, 𝐴, 1) ∈
ℝ+) |
| 45 | 44 | rpne0d 13082 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → if(1 ≤
𝐴, 𝐴, 1) ≠ 0) |
| 46 | 42, 45 | recid2d 12039 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((1 /
if(1 ≤ 𝐴, 𝐴, 1)) · if(1 ≤ 𝐴, 𝐴, 1)) = 1) |
| 47 | 46 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝐵↑𝑐𝑛)↑𝑐((1 /
if(1 ≤ 𝐴, 𝐴, 1)) · if(1 ≤ 𝐴, 𝐴, 1))) = ((𝐵↑𝑐𝑛)↑𝑐1)) |
| 48 | 28, 39, 42 | cxpmuld 26779 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝐵↑𝑐𝑛)↑𝑐((1 /
if(1 ≤ 𝐴, 𝐴, 1)) · if(1 ≤ 𝐴, 𝐴, 1))) = (((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐if(1 ≤
𝐴, 𝐴, 1))) |
| 49 | 28 | rpcnd 13079 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝐵↑𝑐𝑛) ∈
ℂ) |
| 50 | 49 | cxp1d 26748 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝐵↑𝑐𝑛)↑𝑐1) =
(𝐵↑𝑐𝑛)) |
| 51 | 47, 48, 50 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴,
1)))↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) = (𝐵↑𝑐𝑛)) |
| 52 | 51 | oveq2d 7447 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) / (((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐if(1 ≤
𝐴, 𝐴, 1))) = ((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛))) |
| 53 | 43, 52 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1))))↑𝑐if(1 ≤
𝐴, 𝐴, 1)) = ((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛))) |
| 54 | 53 | mpteq2dva 5242 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ ((𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴,
1))))↑𝑐if(1 ≤ 𝐴, 𝐴, 1))) = (𝑛 ∈ ℝ+ ↦ ((𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛)))) |
| 55 | | ovexd 7466 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))) ∈ V) |
| 56 | 18 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 𝑛 ∈
ℂ) |
| 57 | 38 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (1 / if(1 ≤
𝐴, 𝐴, 1)) ∈ ℂ) |
| 58 | 57 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (1 / if(1
≤ 𝐴, 𝐴, 1)) ∈ ℂ) |
| 59 | 56, 58 | mulcomd 11282 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝑛 · (1 / if(1 ≤ 𝐴, 𝐴, 1))) = ((1 / if(1 ≤ 𝐴, 𝐴, 1)) · 𝑛)) |
| 60 | 59 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝐵↑𝑐(𝑛 · (1 / if(1 ≤ 𝐴, 𝐴, 1)))) = (𝐵↑𝑐((1 / if(1 ≤
𝐴, 𝐴, 1)) · 𝑛))) |
| 61 | 27, 18, 58 | cxpmuld 26779 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝐵↑𝑐(𝑛 · (1 / if(1 ≤ 𝐴, 𝐴, 1)))) = ((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))) |
| 62 | 27, 39, 56 | cxpmuld 26779 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝐵↑𝑐((1 /
if(1 ≤ 𝐴, 𝐴, 1)) · 𝑛)) = ((𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐𝑛)) |
| 63 | 60, 61, 62 | 3eqtr3d 2785 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1))) = ((𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐𝑛)) |
| 64 | 63 | oveq2d 7447 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))) = (𝑛 / ((𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐𝑛))) |
| 65 | 64 | mpteq2dva 5242 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ (𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1))))) = (𝑛 ∈ ℝ+ ↦ (𝑛 / ((𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐𝑛)))) |
| 66 | | simp2 1138 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 𝐵 ∈ ℝ) |
| 67 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 1 < 𝐵) |
| 68 | 15, 32, 66, 33, 67 | lttrd 11422 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 0 < 𝐵) |
| 69 | 66, 68 | elrpd 13074 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 𝐵 ∈
ℝ+) |
| 70 | 69, 38 | rpcxpcld 26775 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝐵↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1))) ∈
ℝ+) |
| 71 | 70 | rpred 13077 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝐵↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1))) ∈
ℝ) |
| 72 | 57 | 1cxpd 26749 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) →
(1↑𝑐(1 / if(1 ≤ 𝐴, 𝐴, 1))) = 1) |
| 73 | | 0le1 11786 |
. . . . . . . . . . . . 13
⊢ 0 ≤
1 |
| 74 | 73 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 0 ≤
1) |
| 75 | 69 | rpge0d 13081 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 0 ≤ 𝐵) |
| 76 | 37 | rpreccld 13087 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (1 / if(1 ≤
𝐴, 𝐴, 1)) ∈
ℝ+) |
| 77 | 32, 74, 66, 75, 76 | cxplt2d 26768 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (1 < 𝐵 ↔
(1↑𝑐(1 / if(1 ≤ 𝐴, 𝐴, 1))) < (𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1))))) |
| 78 | 67, 77 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) →
(1↑𝑐(1 / if(1 ≤ 𝐴, 𝐴, 1))) < (𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))) |
| 79 | 72, 78 | eqbrtrrd 5167 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → 1 < (𝐵↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1)))) |
| 80 | | cxp2limlem 27019 |
. . . . . . . . 9
⊢ (((𝐵↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1))) ∈ ℝ ∧ 1
< (𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))) → (𝑛 ∈ ℝ+ ↦ (𝑛 / ((𝐵↑𝑐(1 / if(1 ≤
𝐴, 𝐴, 1)))↑𝑐𝑛))) ⇝𝑟
0) |
| 81 | 71, 79, 80 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ (𝑛 / ((𝐵↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴,
1)))↑𝑐𝑛))) ⇝𝑟
0) |
| 82 | 65, 81 | eqbrtrd 5165 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ (𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴, 1)))))
⇝𝑟 0) |
| 83 | 55, 82, 37 | rlimcxp 27017 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ ((𝑛 / ((𝐵↑𝑐𝑛)↑𝑐(1 /
if(1 ≤ 𝐴, 𝐴,
1))))↑𝑐if(1 ≤ 𝐴, 𝐴, 1))) ⇝𝑟
0) |
| 84 | 54, 83 | eqbrtrrd 5167 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ ((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛))) ⇝𝑟
0) |
| 85 | 16, 84 | rlimres2 15597 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ (1[,)+∞) ↦
((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛))) ⇝𝑟
0) |
| 86 | | simpr 484 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 𝑛 ∈
ℝ+) |
| 87 | 31 | adantr 480 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → if(1 ≤
𝐴, 𝐴, 1) ∈ ℝ) |
| 88 | 86, 87 | rpcxpcld 26775 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) ∈
ℝ+) |
| 89 | 88, 28 | rpdivcld 13094 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛)) ∈
ℝ+) |
| 90 | 89 | rpred 13077 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛)) ∈ ℝ) |
| 91 | 11, 90 | sylan2 593 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → ((𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛)) ∈ ℝ) |
| 92 | | simpl1 1192 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → 𝐴 ∈
ℝ) |
| 93 | 86, 92 | rpcxpcld 26775 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → (𝑛↑𝑐𝐴) ∈
ℝ+) |
| 94 | 93, 28 | rpdivcld 13094 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)) ∈
ℝ+) |
| 95 | 11, 94 | sylan2 593 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)) ∈
ℝ+) |
| 96 | 95 | rpred 13077 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)) ∈ ℝ) |
| 97 | 11, 93 | sylan2 593 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → (𝑛↑𝑐𝐴) ∈
ℝ+) |
| 98 | 97 | rpred 13077 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → (𝑛↑𝑐𝐴) ∈
ℝ) |
| 99 | 11, 88 | sylan2 593 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → (𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) ∈
ℝ+) |
| 100 | 99 | rpred 13077 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → (𝑛↑𝑐if(1
≤ 𝐴, 𝐴, 1)) ∈ ℝ) |
| 101 | 11, 28 | sylan2 593 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → (𝐵↑𝑐𝑛) ∈
ℝ+) |
| 102 | 4 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → 𝑛 ∈
ℝ) |
| 103 | 9 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → 1 ≤ 𝑛) |
| 104 | | simpl1 1192 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → 𝐴 ∈
ℝ) |
| 105 | 31 | adantr 480 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → if(1 ≤
𝐴, 𝐴, 1) ∈ ℝ) |
| 106 | | max2 13229 |
. . . . . . . 8
⊢ ((1
∈ ℝ ∧ 𝐴
∈ ℝ) → 𝐴
≤ if(1 ≤ 𝐴, 𝐴, 1)) |
| 107 | 1, 104, 106 | sylancr 587 |
. . . . . . 7
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → 𝐴 ≤ if(1 ≤ 𝐴, 𝐴, 1)) |
| 108 | 102, 103,
104, 105, 107 | cxplead 26763 |
. . . . . 6
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → (𝑛↑𝑐𝐴) ≤ (𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1))) |
| 109 | 98, 100, 101, 108 | lediv1dd 13135 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)) ≤ ((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛))) |
| 110 | 109 | adantrr 717 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ (𝑛 ∈ (1[,)+∞) ∧ 0
≤ 𝑛)) → ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)) ≤ ((𝑛↑𝑐if(1 ≤ 𝐴, 𝐴, 1)) / (𝐵↑𝑐𝑛))) |
| 111 | 95 | rpge0d 13081 |
. . . . 5
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ (1[,)+∞)) → 0 ≤ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) |
| 112 | 111 | adantrr 717 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ (𝑛 ∈ (1[,)+∞) ∧ 0
≤ 𝑛)) → 0 ≤
((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) |
| 113 | 15, 15, 85, 91, 96, 110, 112 | rlimsqz2 15687 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ (1[,)+∞) ↦
((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ⇝𝑟
0) |
| 114 | 14, 113 | eqbrtrid 5178 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → ((𝑛 ∈ ℝ+
↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ↾ (1[,)+∞))
⇝𝑟 0) |
| 115 | 94 | rpcnd 13079 |
. . . 4
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) ∧ 𝑛 ∈ ℝ+) → ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛)) ∈ ℂ) |
| 116 | 115 | fmpttd 7135 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))):ℝ+⟶ℂ) |
| 117 | | rpssre 13042 |
. . . 4
⊢
ℝ+ ⊆ ℝ |
| 118 | 117 | a1i 11 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) →
ℝ+ ⊆ ℝ) |
| 119 | 116, 118,
32 | rlimresb 15601 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → ((𝑛 ∈ ℝ+
↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ⇝𝑟 0 ↔
((𝑛 ∈
ℝ+ ↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ↾ (1[,)+∞))
⇝𝑟 0)) |
| 120 | 114, 119 | mpbird 257 |
1
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 <
𝐵) → (𝑛 ∈ ℝ+
↦ ((𝑛↑𝑐𝐴) / (𝐵↑𝑐𝑛))) ⇝𝑟
0) |