| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elnn0 12528 | . 2
⊢ (𝐵 ∈ ℕ0
↔ (𝐵 ∈ ℕ
∨ 𝐵 =
0)) | 
| 2 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑧 = 1 → (𝐴↑𝑧) = (𝐴↑1)) | 
| 3 | 2 | eleq1d 2826 | . . . . . 6
⊢ (𝑧 = 1 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑1) ∈ 𝐹)) | 
| 4 | 3 | imbi2d 340 | . . . . 5
⊢ (𝑧 = 1 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑1) ∈ 𝐹))) | 
| 5 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑧 = 𝑤 → (𝐴↑𝑧) = (𝐴↑𝑤)) | 
| 6 | 5 | eleq1d 2826 | . . . . . 6
⊢ (𝑧 = 𝑤 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑𝑤) ∈ 𝐹)) | 
| 7 | 6 | imbi2d 340 | . . . . 5
⊢ (𝑧 = 𝑤 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑𝑤) ∈ 𝐹))) | 
| 8 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑧 = (𝑤 + 1) → (𝐴↑𝑧) = (𝐴↑(𝑤 + 1))) | 
| 9 | 8 | eleq1d 2826 | . . . . . 6
⊢ (𝑧 = (𝑤 + 1) → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑(𝑤 + 1)) ∈ 𝐹)) | 
| 10 | 9 | imbi2d 340 | . . . . 5
⊢ (𝑧 = (𝑤 + 1) → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹))) | 
| 11 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑧 = 𝐵 → (𝐴↑𝑧) = (𝐴↑𝐵)) | 
| 12 | 11 | eleq1d 2826 | . . . . . 6
⊢ (𝑧 = 𝐵 → ((𝐴↑𝑧) ∈ 𝐹 ↔ (𝐴↑𝐵) ∈ 𝐹)) | 
| 13 | 12 | imbi2d 340 | . . . . 5
⊢ (𝑧 = 𝐵 → ((𝐴 ∈ 𝐹 → (𝐴↑𝑧) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹))) | 
| 14 |  | expcllem.1 | . . . . . . . . 9
⊢ 𝐹 ⊆
ℂ | 
| 15 | 14 | sseli 3979 | . . . . . . . 8
⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ ℂ) | 
| 16 |  | exp1 14108 | . . . . . . . 8
⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | 
| 17 | 15, 16 | syl 17 | . . . . . . 7
⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) = 𝐴) | 
| 18 | 17 | eleq1d 2826 | . . . . . 6
⊢ (𝐴 ∈ 𝐹 → ((𝐴↑1) ∈ 𝐹 ↔ 𝐴 ∈ 𝐹)) | 
| 19 | 18 | ibir 268 | . . . . 5
⊢ (𝐴 ∈ 𝐹 → (𝐴↑1) ∈ 𝐹) | 
| 20 |  | expcllem.2 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) | 
| 21 | 20 | caovcl 7627 | . . . . . . . . . . 11
⊢ (((𝐴↑𝑤) ∈ 𝐹 ∧ 𝐴 ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹) | 
| 22 | 21 | ancoms 458 | . . . . . . . . . 10
⊢ ((𝐴 ∈ 𝐹 ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹) | 
| 23 | 22 | adantlr 715 | . . . . . . . . 9
⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑𝑤) · 𝐴) ∈ 𝐹) | 
| 24 |  | nnnn0 12533 | . . . . . . . . . . . 12
⊢ (𝑤 ∈ ℕ → 𝑤 ∈
ℕ0) | 
| 25 |  | expp1 14109 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℕ0)
→ (𝐴↑(𝑤 + 1)) = ((𝐴↑𝑤) · 𝐴)) | 
| 26 | 15, 24, 25 | syl2an 596 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) → (𝐴↑(𝑤 + 1)) = ((𝐴↑𝑤) · 𝐴)) | 
| 27 | 26 | eleq1d 2826 | . . . . . . . . . 10
⊢ ((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) → ((𝐴↑(𝑤 + 1)) ∈ 𝐹 ↔ ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)) | 
| 28 | 27 | adantr 480 | . . . . . . . . 9
⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → ((𝐴↑(𝑤 + 1)) ∈ 𝐹 ↔ ((𝐴↑𝑤) · 𝐴) ∈ 𝐹)) | 
| 29 | 23, 28 | mpbird 257 | . . . . . . . 8
⊢ (((𝐴 ∈ 𝐹 ∧ 𝑤 ∈ ℕ) ∧ (𝐴↑𝑤) ∈ 𝐹) → (𝐴↑(𝑤 + 1)) ∈ 𝐹) | 
| 30 | 29 | exp31 419 | . . . . . . 7
⊢ (𝐴 ∈ 𝐹 → (𝑤 ∈ ℕ → ((𝐴↑𝑤) ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹))) | 
| 31 | 30 | com12 32 | . . . . . 6
⊢ (𝑤 ∈ ℕ → (𝐴 ∈ 𝐹 → ((𝐴↑𝑤) ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹))) | 
| 32 | 31 | a2d 29 | . . . . 5
⊢ (𝑤 ∈ ℕ → ((𝐴 ∈ 𝐹 → (𝐴↑𝑤) ∈ 𝐹) → (𝐴 ∈ 𝐹 → (𝐴↑(𝑤 + 1)) ∈ 𝐹))) | 
| 33 | 4, 7, 10, 13, 19, 32 | nnind 12284 | . . . 4
⊢ (𝐵 ∈ ℕ → (𝐴 ∈ 𝐹 → (𝐴↑𝐵) ∈ 𝐹)) | 
| 34 | 33 | impcom 407 | . . 3
⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ) → (𝐴↑𝐵) ∈ 𝐹) | 
| 35 |  | oveq2 7439 | . . . . 5
⊢ (𝐵 = 0 → (𝐴↑𝐵) = (𝐴↑0)) | 
| 36 |  | exp0 14106 | . . . . . 6
⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | 
| 37 | 15, 36 | syl 17 | . . . . 5
⊢ (𝐴 ∈ 𝐹 → (𝐴↑0) = 1) | 
| 38 | 35, 37 | sylan9eqr 2799 | . . . 4
⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) = 1) | 
| 39 |  | expcllem.3 | . . . 4
⊢ 1 ∈
𝐹 | 
| 40 | 38, 39 | eqeltrdi 2849 | . . 3
⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 = 0) → (𝐴↑𝐵) ∈ 𝐹) | 
| 41 | 34, 40 | jaodan 960 | . 2
⊢ ((𝐴 ∈ 𝐹 ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝐴↑𝐵) ∈ 𝐹) | 
| 42 | 1, 41 | sylan2b 594 | 1
⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ 𝐹) |