Proof of Theorem expubnd
| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) →
𝐴 ∈
ℝ) |
| 2 | | 2re 12340 |
. . . . 5
⊢ 2 ∈
ℝ |
| 3 | | peano2rem 11576 |
. . . . 5
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈
ℝ) |
| 4 | | remulcl 11240 |
. . . . 5
⊢ ((2
∈ ℝ ∧ (𝐴
− 1) ∈ ℝ) → (2 · (𝐴 − 1)) ∈
ℝ) |
| 5 | 2, 3, 4 | sylancr 587 |
. . . 4
⊢ (𝐴 ∈ ℝ → (2
· (𝐴 − 1))
∈ ℝ) |
| 6 | 5 | 3ad2ant1 1134 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) → (2
· (𝐴 − 1))
∈ ℝ) |
| 7 | | simp2 1138 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) →
𝑁 ∈
ℕ0) |
| 8 | | 0le2 12368 |
. . . . . . 7
⊢ 0 ≤
2 |
| 9 | | 0re 11263 |
. . . . . . . 8
⊢ 0 ∈
ℝ |
| 10 | | letr 11355 |
. . . . . . . 8
⊢ ((0
∈ ℝ ∧ 2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 ≤ 2 ∧ 2
≤ 𝐴) → 0 ≤ 𝐴)) |
| 11 | 9, 2, 10 | mp3an12 1453 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → ((0 ≤
2 ∧ 2 ≤ 𝐴) → 0
≤ 𝐴)) |
| 12 | 8, 11 | mpani 696 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → (2 ≤
𝐴 → 0 ≤ 𝐴)) |
| 13 | 12 | imp 406 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤
𝐴) → 0 ≤ 𝐴) |
| 14 | | resubcl 11573 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 2 ∈
ℝ) → (𝐴 −
2) ∈ ℝ) |
| 15 | 2, 14 | mpan2 691 |
. . . . . . . 8
⊢ (𝐴 ∈ ℝ → (𝐴 − 2) ∈
ℝ) |
| 16 | | leadd2 11732 |
. . . . . . . . 9
⊢ ((2
∈ ℝ ∧ 𝐴
∈ ℝ ∧ (𝐴
− 2) ∈ ℝ) → (2 ≤ 𝐴 ↔ ((𝐴 − 2) + 2) ≤ ((𝐴 − 2) + 𝐴))) |
| 17 | 2, 16 | mp3an1 1450 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ (𝐴 − 2) ∈ ℝ)
→ (2 ≤ 𝐴 ↔
((𝐴 − 2) + 2) ≤
((𝐴 − 2) + 𝐴))) |
| 18 | 15, 17 | mpdan 687 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → (2 ≤
𝐴 ↔ ((𝐴 − 2) + 2) ≤ ((𝐴 − 2) + 𝐴))) |
| 19 | 18 | biimpa 476 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤
𝐴) → ((𝐴 − 2) + 2) ≤ ((𝐴 − 2) + 𝐴)) |
| 20 | | recn 11245 |
. . . . . . . 8
⊢ (𝐴 ∈ ℝ → 𝐴 ∈
ℂ) |
| 21 | | 2cn 12341 |
. . . . . . . 8
⊢ 2 ∈
ℂ |
| 22 | | npcan 11517 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 2 ∈
ℂ) → ((𝐴 −
2) + 2) = 𝐴) |
| 23 | 20, 21, 22 | sylancl 586 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → ((𝐴 − 2) + 2) = 𝐴) |
| 24 | 23 | adantr 480 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤
𝐴) → ((𝐴 − 2) + 2) = 𝐴) |
| 25 | | ax-1cn 11213 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
| 26 | | subdi 11696 |
. . . . . . . . . 10
⊢ ((2
∈ ℂ ∧ 𝐴
∈ ℂ ∧ 1 ∈ ℂ) → (2 · (𝐴 − 1)) = ((2 · 𝐴) − (2 ·
1))) |
| 27 | 21, 25, 26 | mp3an13 1454 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℂ → (2
· (𝐴 − 1)) =
((2 · 𝐴) − (2
· 1))) |
| 28 | | 2times 12402 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℂ → (2
· 𝐴) = (𝐴 + 𝐴)) |
| 29 | | 2t1e2 12429 |
. . . . . . . . . . 11
⊢ (2
· 1) = 2 |
| 30 | 29 | a1i 11 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℂ → (2
· 1) = 2) |
| 31 | 28, 30 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℂ → ((2
· 𝐴) − (2
· 1)) = ((𝐴 + 𝐴) − 2)) |
| 32 | | addsub 11519 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 2 ∈
ℂ) → ((𝐴 + 𝐴) − 2) = ((𝐴 − 2) + 𝐴)) |
| 33 | 21, 32 | mp3an3 1452 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐴) − 2) = ((𝐴 − 2) + 𝐴)) |
| 34 | 33 | anidms 566 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) − 2) = ((𝐴 − 2) + 𝐴)) |
| 35 | 27, 31, 34 | 3eqtrrd 2782 |
. . . . . . . 8
⊢ (𝐴 ∈ ℂ → ((𝐴 − 2) + 𝐴) = (2 · (𝐴 − 1))) |
| 36 | 20, 35 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → ((𝐴 − 2) + 𝐴) = (2 · (𝐴 − 1))) |
| 37 | 36 | adantr 480 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤
𝐴) → ((𝐴 − 2) + 𝐴) = (2 · (𝐴 − 1))) |
| 38 | 19, 24, 37 | 3brtr3d 5174 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤
𝐴) → 𝐴 ≤ (2 · (𝐴 − 1))) |
| 39 | 13, 38 | jca 511 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 2 ≤
𝐴) → (0 ≤ 𝐴 ∧ 𝐴 ≤ (2 · (𝐴 − 1)))) |
| 40 | 39 | 3adant2 1132 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) → (0
≤ 𝐴 ∧ 𝐴 ≤ (2 · (𝐴 − 1)))) |
| 41 | | leexp1a 14215 |
. . 3
⊢ (((𝐴 ∈ ℝ ∧ (2
· (𝐴 − 1))
∈ ℝ ∧ 𝑁
∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ (2 · (𝐴 − 1)))) → (𝐴↑𝑁) ≤ ((2 · (𝐴 − 1))↑𝑁)) |
| 42 | 1, 6, 7, 40, 41 | syl31anc 1375 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) →
(𝐴↑𝑁) ≤ ((2 · (𝐴 − 1))↑𝑁)) |
| 43 | 3 | recnd 11289 |
. . . 4
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈
ℂ) |
| 44 | | mulexp 14142 |
. . . . 5
⊢ ((2
∈ ℂ ∧ (𝐴
− 1) ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((2
· (𝐴 −
1))↑𝑁) =
((2↑𝑁) ·
((𝐴 − 1)↑𝑁))) |
| 45 | 21, 44 | mp3an1 1450 |
. . . 4
⊢ (((𝐴 − 1) ∈ ℂ ∧
𝑁 ∈
ℕ0) → ((2 · (𝐴 − 1))↑𝑁) = ((2↑𝑁) · ((𝐴 − 1)↑𝑁))) |
| 46 | 43, 45 | sylan 580 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0)
→ ((2 · (𝐴
− 1))↑𝑁) =
((2↑𝑁) ·
((𝐴 − 1)↑𝑁))) |
| 47 | 46 | 3adant3 1133 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) → ((2
· (𝐴 −
1))↑𝑁) =
((2↑𝑁) ·
((𝐴 − 1)↑𝑁))) |
| 48 | 42, 47 | breqtrd 5169 |
1
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0
∧ 2 ≤ 𝐴) →
(𝐴↑𝑁) ≤ ((2↑𝑁) · ((𝐴 − 1)↑𝑁))) |