| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . 3
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ∈
ℕ) |
| 2 | | simpl2 1193 |
. . 3
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐵 ∈
ℝ) |
| 3 | | simpl3 1194 |
. . 3
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 2 ≤
𝐵) |
| 4 | | id 22 |
. . . . . . 7
⊢ (𝑘 = 1 → 𝑘 = 1) |
| 5 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑘 = 1 → (𝐵↑𝑘) = (𝐵↑1)) |
| 6 | 4, 5 | breq12d 5156 |
. . . . . 6
⊢ (𝑘 = 1 → (𝑘 ≤ (𝐵↑𝑘) ↔ 1 ≤ (𝐵↑1))) |
| 7 | 6 | imbi2d 340 |
. . . . 5
⊢ (𝑘 = 1 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 1 ≤ (𝐵↑1)))) |
| 8 | | id 22 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛) |
| 9 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → (𝐵↑𝑘) = (𝐵↑𝑛)) |
| 10 | 8, 9 | breq12d 5156 |
. . . . . 6
⊢ (𝑘 = 𝑛 → (𝑘 ≤ (𝐵↑𝑘) ↔ 𝑛 ≤ (𝐵↑𝑛))) |
| 11 | 10 | imbi2d 340 |
. . . . 5
⊢ (𝑘 = 𝑛 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑛 ≤ (𝐵↑𝑛)))) |
| 12 | | id 22 |
. . . . . . 7
⊢ (𝑘 = (𝑛 + 1) → 𝑘 = (𝑛 + 1)) |
| 13 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑘 = (𝑛 + 1) → (𝐵↑𝑘) = (𝐵↑(𝑛 + 1))) |
| 14 | 12, 13 | breq12d 5156 |
. . . . . 6
⊢ (𝑘 = (𝑛 + 1) → (𝑘 ≤ (𝐵↑𝑘) ↔ (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1)))) |
| 15 | 14 | imbi2d 340 |
. . . . 5
⊢ (𝑘 = (𝑛 + 1) → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))) |
| 16 | | id 22 |
. . . . . . 7
⊢ (𝑘 = 𝐴 → 𝑘 = 𝐴) |
| 17 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑘 = 𝐴 → (𝐵↑𝑘) = (𝐵↑𝐴)) |
| 18 | 16, 17 | breq12d 5156 |
. . . . . 6
⊢ (𝑘 = 𝐴 → (𝑘 ≤ (𝐵↑𝑘) ↔ 𝐴 ≤ (𝐵↑𝐴))) |
| 19 | 18 | imbi2d 340 |
. . . . 5
⊢ (𝑘 = 𝐴 → (((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝑘 ≤ (𝐵↑𝑘)) ↔ ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴)))) |
| 20 | | simpl 482 |
. . . . . 6
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 𝐵 ∈ ℝ) |
| 21 | | 1nn0 12542 |
. . . . . . 7
⊢ 1 ∈
ℕ0 |
| 22 | 21 | a1i 11 |
. . . . . 6
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 1 ∈
ℕ0) |
| 23 | | 1red 11262 |
. . . . . . 7
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 1 ∈
ℝ) |
| 24 | | 2re 12340 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
| 25 | 24 | a1i 11 |
. . . . . . 7
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 2 ∈
ℝ) |
| 26 | | 1le2 12475 |
. . . . . . . 8
⊢ 1 ≤
2 |
| 27 | 26 | a1i 11 |
. . . . . . 7
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 1 ≤
2) |
| 28 | | simpr 484 |
. . . . . . 7
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 2 ≤ 𝐵) |
| 29 | 23, 25, 20, 27, 28 | letrd 11418 |
. . . . . 6
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 1 ≤ 𝐵) |
| 30 | 20, 22, 29 | expge1d 14205 |
. . . . 5
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 1 ≤ (𝐵↑1)) |
| 31 | | simp1 1137 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ) |
| 32 | 31 | nnred 12281 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℝ) |
| 33 | | 1red 11262 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ∈ ℝ) |
| 34 | 32, 33 | readdcld 11290 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ∈ ℝ) |
| 35 | 20 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℝ) |
| 36 | 32, 35 | remulcld 11291 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ∈ ℝ) |
| 37 | 31 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℕ0) |
| 38 | 35, 37 | reexpcld 14203 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑𝑛) ∈ ℝ) |
| 39 | 38, 35 | remulcld 11291 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → ((𝐵↑𝑛) · 𝐵) ∈ ℝ) |
| 40 | 24 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 2 ∈ ℝ) |
| 41 | 32, 40 | remulcld 11291 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ∈ ℝ) |
| 42 | 31 | nnge1d 12314 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 1 ≤ 𝑛) |
| 43 | 33, 32, 32, 42 | leadd2dd 11878 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 + 𝑛)) |
| 44 | 32 | recnd 11289 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ∈ ℂ) |
| 45 | 44 | times2d 12510 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) = (𝑛 + 𝑛)) |
| 46 | 43, 45 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 2)) |
| 47 | 37 | nn0ge0d 12590 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 0 ≤ 𝑛) |
| 48 | | simp2r 1201 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 2 ≤ 𝐵) |
| 49 | 40, 35, 32, 47, 48 | lemul2ad 12208 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 2) ≤ (𝑛 · 𝐵)) |
| 50 | 34, 41, 36, 46, 49 | letrd 11418 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝑛 · 𝐵)) |
| 51 | | 0red 11264 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 0 ∈
ℝ) |
| 52 | | 0le2 12368 |
. . . . . . . . . . . . 13
⊢ 0 ≤
2 |
| 53 | 52 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 0 ≤
2) |
| 54 | 51, 25, 20, 53, 28 | letrd 11418 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 0 ≤ 𝐵) |
| 55 | 54 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 0 ≤ 𝐵) |
| 56 | | simp3 1139 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝑛 ≤ (𝐵↑𝑛)) |
| 57 | 32, 38, 35, 55, 56 | lemul1ad 12207 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 · 𝐵) ≤ ((𝐵↑𝑛) · 𝐵)) |
| 58 | 34, 36, 39, 50, 57 | letrd 11418 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ ((𝐵↑𝑛) · 𝐵)) |
| 59 | 35 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → 𝐵 ∈ ℂ) |
| 60 | 59, 37 | expp1d 14187 |
. . . . . . . 8
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝐵↑(𝑛 + 1)) = ((𝐵↑𝑛) · 𝐵)) |
| 61 | 58, 60 | breqtrrd 5171 |
. . . . . . 7
⊢ ((𝑛 ∈ ℕ ∧ (𝐵 ∈ ℝ ∧ 2 ≤
𝐵) ∧ 𝑛 ≤ (𝐵↑𝑛)) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))) |
| 62 | 61 | 3exp 1120 |
. . . . . 6
⊢ (𝑛 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → (𝑛 ≤ (𝐵↑𝑛) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))) |
| 63 | 62 | a2d 29 |
. . . . 5
⊢ (𝑛 ∈ ℕ → (((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 𝑛 ≤ (𝐵↑𝑛)) → ((𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → (𝑛 + 1) ≤ (𝐵↑(𝑛 + 1))))) |
| 64 | 7, 11, 15, 19, 30, 63 | nnind 12284 |
. . . 4
⊢ (𝐴 ∈ ℕ → ((𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 𝐴 ≤ (𝐵↑𝐴))) |
| 65 | 64 | 3impib 1117 |
. . 3
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ ∧ 2 ≤
𝐵) → 𝐴 ≤ (𝐵↑𝐴)) |
| 66 | 1, 2, 3, 65 | syl3anc 1373 |
. 2
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 ∈ ℕ) → 𝐴 ≤ (𝐵↑𝐴)) |
| 67 | | 0le1 11786 |
. . . 4
⊢ 0 ≤
1 |
| 68 | 67 | a1i 11 |
. . 3
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 0 ≤
1) |
| 69 | | simpr 484 |
. . 3
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 = 0) |
| 70 | 69 | oveq2d 7447 |
. . . 4
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = (𝐵↑0)) |
| 71 | | simpl2 1193 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℝ) |
| 72 | 71 | recnd 11289 |
. . . . 5
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐵 ∈ ℂ) |
| 73 | 72 | exp0d 14180 |
. . . 4
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑0) = 1) |
| 74 | 70, 73 | eqtrd 2777 |
. . 3
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → (𝐵↑𝐴) = 1) |
| 75 | 68, 69, 74 | 3brtr4d 5175 |
. 2
⊢ (((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) ∧ 𝐴 = 0) → 𝐴 ≤ (𝐵↑𝐴)) |
| 76 | | elnn0 12528 |
. . . 4
⊢ (𝐴 ∈ ℕ0
↔ (𝐴 ∈ ℕ
∨ 𝐴 =
0)) |
| 77 | 76 | biimpi 216 |
. . 3
⊢ (𝐴 ∈ ℕ0
→ (𝐴 ∈ ℕ
∨ 𝐴 =
0)) |
| 78 | 77 | 3ad2ant1 1134 |
. 2
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) →
(𝐴 ∈ ℕ ∨
𝐴 = 0)) |
| 79 | 66, 75, 78 | mpjaodan 961 |
1
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈ ℝ
∧ 2 ≤ 𝐵) →
𝐴 ≤ (𝐵↑𝐴)) |