Proof of Theorem expgt0b
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | expgt0b.n | . . . . 5
⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 2 | 1 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℝ) | 
| 3 |  | expgt0b.m | . . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 4 | 3 | nnzd 12640 | . . . . 5
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 5 | 4 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝑁 ∈ ℤ) | 
| 6 |  | simpr 484 | . . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < 𝐴) | 
| 7 |  | expgt0 14136 | . . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 <
𝐴) → 0 < (𝐴↑𝑁)) | 
| 8 | 2, 5, 6, 7 | syl3anc 1373 | . . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → 0 < (𝐴↑𝑁)) | 
| 9 | 8 | ex 412 | . 2
⊢ (𝜑 → (0 < 𝐴 → 0 < (𝐴↑𝑁))) | 
| 10 |  | 0red 11264 | . . . . 5
⊢ (𝜑 → 0 ∈
ℝ) | 
| 11 | 10, 1 | lttrid 11399 | . . . 4
⊢ (𝜑 → (0 < 𝐴 ↔ ¬ (0 = 𝐴 ∨ 𝐴 < 0))) | 
| 12 | 11 | notbid 318 | . . 3
⊢ (𝜑 → (¬ 0 < 𝐴 ↔ ¬ ¬ (0 = 𝐴 ∨ 𝐴 < 0))) | 
| 13 |  | notnotr 130 | . . . 4
⊢ (¬
¬ (0 = 𝐴 ∨ 𝐴 < 0) → (0 = 𝐴 ∨ 𝐴 < 0)) | 
| 14 |  | 0re 11263 | . . . . . . . . . 10
⊢ 0 ∈
ℝ | 
| 15 | 14 | ltnri 11370 | . . . . . . . . 9
⊢  ¬ 0
< 0 | 
| 16 | 3 | 0expd 14179 | . . . . . . . . . 10
⊢ (𝜑 → (0↑𝑁) = 0) | 
| 17 | 16 | breq2d 5155 | . . . . . . . . 9
⊢ (𝜑 → (0 < (0↑𝑁) ↔ 0 <
0)) | 
| 18 | 15, 17 | mtbiri 327 | . . . . . . . 8
⊢ (𝜑 → ¬ 0 < (0↑𝑁)) | 
| 19 | 18 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 0 = 𝐴) → ¬ 0 < (0↑𝑁)) | 
| 20 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 0 = 𝐴) → 0 = 𝐴) | 
| 21 | 20 | eqcomd 2743 | . . . . . . . . 9
⊢ ((𝜑 ∧ 0 = 𝐴) → 𝐴 = 0) | 
| 22 | 21 | oveq1d 7446 | . . . . . . . 8
⊢ ((𝜑 ∧ 0 = 𝐴) → (𝐴↑𝑁) = (0↑𝑁)) | 
| 23 | 22 | breq2d 5155 | . . . . . . 7
⊢ ((𝜑 ∧ 0 = 𝐴) → (0 < (𝐴↑𝑁) ↔ 0 < (0↑𝑁))) | 
| 24 | 19, 23 | mtbird 325 | . . . . . 6
⊢ ((𝜑 ∧ 0 = 𝐴) → ¬ 0 < (𝐴↑𝑁)) | 
| 25 | 24 | ex 412 | . . . . 5
⊢ (𝜑 → (0 = 𝐴 → ¬ 0 < (𝐴↑𝑁))) | 
| 26 | 1 | renegcld 11690 | . . . . . . . . . 10
⊢ (𝜑 → -𝐴 ∈ ℝ) | 
| 27 | 26 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 0 < -𝐴) → -𝐴 ∈ ℝ) | 
| 28 | 4 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 0 < -𝐴) → 𝑁 ∈ ℤ) | 
| 29 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝜑 ∧ 0 < -𝐴) → 0 < -𝐴) | 
| 30 |  | expgt0 14136 | . . . . . . . . 9
⊢ ((-𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 <
-𝐴) → 0 < (-𝐴↑𝑁)) | 
| 31 | 27, 28, 29, 30 | syl3anc 1373 | . . . . . . . 8
⊢ ((𝜑 ∧ 0 < -𝐴) → 0 < (-𝐴↑𝑁)) | 
| 32 | 31 | ex 412 | . . . . . . 7
⊢ (𝜑 → (0 < -𝐴 → 0 < (-𝐴↑𝑁))) | 
| 33 | 1 | recnd 11289 | . . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| 34 |  | expgt0b.1 | . . . . . . . . . 10
⊢ (𝜑 → ¬ 2 ∥ 𝑁) | 
| 35 |  | oexpneg 16382 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ ¬ 2
∥ 𝑁) → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | 
| 36 | 33, 3, 34, 35 | syl3anc 1373 | . . . . . . . . 9
⊢ (𝜑 → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | 
| 37 | 36 | breq2d 5155 | . . . . . . . 8
⊢ (𝜑 → (0 < (-𝐴↑𝑁) ↔ 0 < -(𝐴↑𝑁))) | 
| 38 | 37 | biimpd 229 | . . . . . . 7
⊢ (𝜑 → (0 < (-𝐴↑𝑁) → 0 < -(𝐴↑𝑁))) | 
| 39 | 3 | nnnn0d 12587 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) | 
| 40 | 1, 39 | reexpcld 14203 | . . . . . . . . . 10
⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | 
| 41 | 40 | renegcld 11690 | . . . . . . . . 9
⊢ (𝜑 → -(𝐴↑𝑁) ∈ ℝ) | 
| 42 | 10, 41 | lttrid 11399 | . . . . . . . 8
⊢ (𝜑 → (0 < -(𝐴↑𝑁) ↔ ¬ (0 = -(𝐴↑𝑁) ∨ -(𝐴↑𝑁) < 0))) | 
| 43 |  | pm2.46 883 | . . . . . . . 8
⊢ (¬ (0
= -(𝐴↑𝑁) ∨ -(𝐴↑𝑁) < 0) → ¬ -(𝐴↑𝑁) < 0) | 
| 44 | 42, 43 | biimtrdi 253 | . . . . . . 7
⊢ (𝜑 → (0 < -(𝐴↑𝑁) → ¬ -(𝐴↑𝑁) < 0)) | 
| 45 | 32, 38, 44 | 3syld 60 | . . . . . 6
⊢ (𝜑 → (0 < -𝐴 → ¬ -(𝐴↑𝑁) < 0)) | 
| 46 | 1 | lt0neg1d 11832 | . . . . . 6
⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) | 
| 47 | 40 | lt0neg2d 11833 | . . . . . . 7
⊢ (𝜑 → (0 < (𝐴↑𝑁) ↔ -(𝐴↑𝑁) < 0)) | 
| 48 | 47 | notbid 318 | . . . . . 6
⊢ (𝜑 → (¬ 0 < (𝐴↑𝑁) ↔ ¬ -(𝐴↑𝑁) < 0)) | 
| 49 | 45, 46, 48 | 3imtr4d 294 | . . . . 5
⊢ (𝜑 → (𝐴 < 0 → ¬ 0 < (𝐴↑𝑁))) | 
| 50 | 25, 49 | jaod 860 | . . . 4
⊢ (𝜑 → ((0 = 𝐴 ∨ 𝐴 < 0) → ¬ 0 < (𝐴↑𝑁))) | 
| 51 | 13, 50 | syl5 34 | . . 3
⊢ (𝜑 → (¬ ¬ (0 = 𝐴 ∨ 𝐴 < 0) → ¬ 0 < (𝐴↑𝑁))) | 
| 52 | 12, 51 | sylbid 240 | . 2
⊢ (𝜑 → (¬ 0 < 𝐴 → ¬ 0 < (𝐴↑𝑁))) | 
| 53 | 9, 52 | impcon4bid 227 | 1
⊢ (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴↑𝑁))) |