| Step | Hyp | Ref
| Expression |
| 1 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = 0 → (𝐵 · 𝑥) = (𝐵 · 0)) |
| 2 | 1 | oveq2d 7447 |
. . . . . 6
⊢ (𝑥 = 0 → (𝐴↑𝑐(𝐵 · 𝑥)) = (𝐴↑𝑐(𝐵 · 0))) |
| 3 | | oveq2 7439 |
. . . . . 6
⊢ (𝑥 = 0 → ((𝐴↑𝑐𝐵)↑𝑥) = ((𝐴↑𝑐𝐵)↑0)) |
| 4 | 2, 3 | eqeq12d 2753 |
. . . . 5
⊢ (𝑥 = 0 → ((𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥) ↔ (𝐴↑𝑐(𝐵 · 0)) = ((𝐴↑𝑐𝐵)↑0))) |
| 5 | 4 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 0 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥)) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 0)) = ((𝐴↑𝑐𝐵)↑0)))) |
| 6 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = 𝑘 → (𝐵 · 𝑥) = (𝐵 · 𝑘)) |
| 7 | 6 | oveq2d 7447 |
. . . . . 6
⊢ (𝑥 = 𝑘 → (𝐴↑𝑐(𝐵 · 𝑥)) = (𝐴↑𝑐(𝐵 · 𝑘))) |
| 8 | | oveq2 7439 |
. . . . . 6
⊢ (𝑥 = 𝑘 → ((𝐴↑𝑐𝐵)↑𝑥) = ((𝐴↑𝑐𝐵)↑𝑘)) |
| 9 | 7, 8 | eqeq12d 2753 |
. . . . 5
⊢ (𝑥 = 𝑘 → ((𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥) ↔ (𝐴↑𝑐(𝐵 · 𝑘)) = ((𝐴↑𝑐𝐵)↑𝑘))) |
| 10 | 9 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝑘 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥)) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝑘)) = ((𝐴↑𝑐𝐵)↑𝑘)))) |
| 11 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = (𝑘 + 1) → (𝐵 · 𝑥) = (𝐵 · (𝑘 + 1))) |
| 12 | 11 | oveq2d 7447 |
. . . . . 6
⊢ (𝑥 = (𝑘 + 1) → (𝐴↑𝑐(𝐵 · 𝑥)) = (𝐴↑𝑐(𝐵 · (𝑘 + 1)))) |
| 13 | | oveq2 7439 |
. . . . . 6
⊢ (𝑥 = (𝑘 + 1) → ((𝐴↑𝑐𝐵)↑𝑥) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1))) |
| 14 | 12, 13 | eqeq12d 2753 |
. . . . 5
⊢ (𝑥 = (𝑘 + 1) → ((𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥) ↔ (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1)))) |
| 15 | 14 | imbi2d 340 |
. . . 4
⊢ (𝑥 = (𝑘 + 1) → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥)) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1))))) |
| 16 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐵 · 𝑥) = (𝐵 · 𝐶)) |
| 17 | 16 | oveq2d 7447 |
. . . . . 6
⊢ (𝑥 = 𝐶 → (𝐴↑𝑐(𝐵 · 𝑥)) = (𝐴↑𝑐(𝐵 · 𝐶))) |
| 18 | | oveq2 7439 |
. . . . . 6
⊢ (𝑥 = 𝐶 → ((𝐴↑𝑐𝐵)↑𝑥) = ((𝐴↑𝑐𝐵)↑𝐶)) |
| 19 | 17, 18 | eqeq12d 2753 |
. . . . 5
⊢ (𝑥 = 𝐶 → ((𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥) ↔ (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))) |
| 20 | 19 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝑥)) = ((𝐴↑𝑐𝐵)↑𝑥)) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶)))) |
| 21 | | cxp0 26712 |
. . . . . 6
⊢ (𝐴 ∈ ℂ → (𝐴↑𝑐0) =
1) |
| 22 | 21 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐0) =
1) |
| 23 | | mul01 11440 |
. . . . . . 7
⊢ (𝐵 ∈ ℂ → (𝐵 · 0) =
0) |
| 24 | 23 | adantl 481 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 · 0) =
0) |
| 25 | 24 | oveq2d 7447 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 0)) = (𝐴↑𝑐0)) |
| 26 | | cxpcl 26716 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) ∈
ℂ) |
| 27 | 26 | exp0d 14180 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑐𝐵)↑0) = 1) |
| 28 | 22, 25, 27 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · 0)) = ((𝐴↑𝑐𝐵)↑0)) |
| 29 | | oveq1 7438 |
. . . . . . 7
⊢ ((𝐴↑𝑐(𝐵 · 𝑘)) = ((𝐴↑𝑐𝐵)↑𝑘) → ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵)) = (((𝐴↑𝑐𝐵)↑𝑘) · (𝐴↑𝑐𝐵))) |
| 30 | | 0cn 11253 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℂ |
| 31 | | cxp0 26712 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℂ → (0↑𝑐0) = 1) |
| 32 | 30, 31 | ax-mp 5 |
. . . . . . . . . . . 12
⊢
(0↑𝑐0) = 1 |
| 33 | | 1t1e1 12428 |
. . . . . . . . . . . 12
⊢ (1
· 1) = 1 |
| 34 | 32, 33 | eqtr4i 2768 |
. . . . . . . . . . 11
⊢
(0↑𝑐0) = (1 · 1) |
| 35 | | simplr 769 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐴 = 0) |
| 36 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐵 = 0) |
| 37 | 36 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐵 · (𝑘 + 1)) = (0 · (𝑘 + 1))) |
| 38 | | nn0p1nn 12565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ) |
| 39 | 38 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ (𝑘 + 1) ∈
ℕ) |
| 40 | 39 | nncnd 12282 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ (𝑘 + 1) ∈
ℂ) |
| 41 | 40 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝑘 + 1) ∈ ℂ) |
| 42 | 41 | mul02d 11459 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (0 · (𝑘 + 1)) = 0) |
| 43 | 37, 42 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐵 · (𝑘 + 1)) = 0) |
| 44 | 35, 43 | oveq12d 7449 |
. . . . . . . . . . 11
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) =
(0↑𝑐0)) |
| 45 | 36 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐵 · 𝑘) = (0 · 𝑘)) |
| 46 | | nn0cn 12536 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℂ) |
| 47 | 46 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ 𝑘 ∈
ℂ) |
| 48 | 47 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝑘 ∈ ℂ) |
| 49 | 48 | mul02d 11459 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (0 · 𝑘) = 0) |
| 50 | 45, 49 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐵 · 𝑘) = 0) |
| 51 | 35, 50 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑𝑐(𝐵 · 𝑘)) =
(0↑𝑐0)) |
| 52 | 51, 32 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑𝑐(𝐵 · 𝑘)) = 1) |
| 53 | 35, 36 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑𝑐𝐵) =
(0↑𝑐0)) |
| 54 | 53, 32 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑𝑐𝐵) = 1) |
| 55 | 52, 54 | oveq12d 7449 |
. . . . . . . . . . 11
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵)) = (1 · 1)) |
| 56 | 34, 44, 55 | 3eqtr4a 2803 |
. . . . . . . . . 10
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 57 | | simpll 767 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ 𝐴 ∈
ℂ) |
| 58 | 57 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐴 ∈ ℂ) |
| 59 | | simplr 769 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ 𝐵 ∈
ℂ) |
| 60 | 59, 47 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ (𝐵 · 𝑘) ∈
ℂ) |
| 61 | 60 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐵 · 𝑘) ∈ ℂ) |
| 62 | | cxpcl 26716 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝑘) ∈ ℂ) → (𝐴↑𝑐(𝐵 · 𝑘)) ∈ ℂ) |
| 63 | 58, 61, 62 | syl2anc 584 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑𝑐(𝐵 · 𝑘)) ∈ ℂ) |
| 64 | 63 | mul01d 11460 |
. . . . . . . . . . 11
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((𝐴↑𝑐(𝐵 · 𝑘)) · 0) = 0) |
| 65 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐴 = 0) |
| 66 | 65 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑𝑐𝐵) = (0↑𝑐𝐵)) |
| 67 | 59 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ) |
| 68 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) |
| 69 | | 0cxp 26708 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) →
(0↑𝑐𝐵) = 0) |
| 70 | 67, 68, 69 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) →
(0↑𝑐𝐵) = 0) |
| 71 | 66, 70 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑𝑐𝐵) = 0) |
| 72 | 71 | oveq2d 7447 |
. . . . . . . . . . 11
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵)) = ((𝐴↑𝑐(𝐵 · 𝑘)) · 0)) |
| 73 | 65 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = (0↑𝑐(𝐵 · (𝑘 + 1)))) |
| 74 | 40 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝑘 + 1) ∈ ℂ) |
| 75 | 67, 74 | mulcld 11281 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐵 · (𝑘 + 1)) ∈ ℂ) |
| 76 | 39 | nnne0d 12316 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ (𝑘 + 1) ≠
0) |
| 77 | 76 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝑘 + 1) ≠ 0) |
| 78 | 67, 74, 68, 77 | mulne0d 11915 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐵 · (𝑘 + 1)) ≠ 0) |
| 79 | | 0cxp 26708 |
. . . . . . . . . . . . 13
⊢ (((𝐵 · (𝑘 + 1)) ∈ ℂ ∧ (𝐵 · (𝑘 + 1)) ≠ 0) →
(0↑𝑐(𝐵 · (𝑘 + 1))) = 0) |
| 80 | 75, 78, 79 | syl2anc 584 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) →
(0↑𝑐(𝐵 · (𝑘 + 1))) = 0) |
| 81 | 73, 80 | eqtrd 2777 |
. . . . . . . . . . 11
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = 0) |
| 82 | 64, 72, 81 | 3eqtr4rd 2788 |
. . . . . . . . . 10
⊢
(((((𝐴 ∈
ℂ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
ℕ0) ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 83 | 56, 82 | pm2.61dane 3029 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 = 0) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 84 | 59 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
𝐵 ∈
ℂ) |
| 85 | 47 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
𝑘 ∈
ℂ) |
| 86 | | 1cnd 11256 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) → 1
∈ ℂ) |
| 87 | 84, 85, 86 | adddid 11285 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐵 · (𝑘 + 1)) = ((𝐵 · 𝑘) + (𝐵 · 1))) |
| 88 | 84 | mulridd 11278 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐵 · 1) = 𝐵) |
| 89 | 88 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
((𝐵 · 𝑘) + (𝐵 · 1)) = ((𝐵 · 𝑘) + 𝐵)) |
| 90 | 87, 89 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐵 · (𝑘 + 1)) = ((𝐵 · 𝑘) + 𝐵)) |
| 91 | 90 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐴↑𝑐(𝐵 · (𝑘 + 1))) = (𝐴↑𝑐((𝐵 · 𝑘) + 𝐵))) |
| 92 | 57 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
𝐴 ∈
ℂ) |
| 93 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
𝐴 ≠ 0) |
| 94 | 60 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐵 · 𝑘) ∈
ℂ) |
| 95 | | cxpadd 26721 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 · 𝑘) ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐((𝐵 · 𝑘) + 𝐵)) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 96 | 92, 93, 94, 84, 95 | syl211anc 1378 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐴↑𝑐((𝐵 · 𝑘) + 𝐵)) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 97 | 91, 96 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
∧ 𝐴 ≠ 0) →
(𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 98 | 83, 97 | pm2.61dane 3029 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵))) |
| 99 | | expp1 14109 |
. . . . . . . . 9
⊢ (((𝐴↑𝑐𝐵) ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ ((𝐴↑𝑐𝐵)↑(𝑘 + 1)) = (((𝐴↑𝑐𝐵)↑𝑘) · (𝐴↑𝑐𝐵))) |
| 100 | 26, 99 | sylan 580 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ ((𝐴↑𝑐𝐵)↑(𝑘 + 1)) = (((𝐴↑𝑐𝐵)↑𝑘) · (𝐴↑𝑐𝐵))) |
| 101 | 98, 100 | eqeq12d 2753 |
. . . . . . 7
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ ((𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1)) ↔ ((𝐴↑𝑐(𝐵 · 𝑘)) · (𝐴↑𝑐𝐵)) = (((𝐴↑𝑐𝐵)↑𝑘) · (𝐴↑𝑐𝐵)))) |
| 102 | 29, 101 | imbitrrid 246 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑘 ∈ ℕ0)
→ ((𝐴↑𝑐(𝐵 · 𝑘)) = ((𝐴↑𝑐𝐵)↑𝑘) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1)))) |
| 103 | 102 | expcom 413 |
. . . . 5
⊢ (𝑘 ∈ ℕ0
→ ((𝐴 ∈ ℂ
∧ 𝐵 ∈ ℂ)
→ ((𝐴↑𝑐(𝐵 · 𝑘)) = ((𝐴↑𝑐𝐵)↑𝑘) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1))))) |
| 104 | 103 | a2d 29 |
. . . 4
⊢ (𝑘 ∈ ℕ0
→ (((𝐴 ∈ ℂ
∧ 𝐵 ∈ ℂ)
→ (𝐴↑𝑐(𝐵 · 𝑘)) = ((𝐴↑𝑐𝐵)↑𝑘)) → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐(𝐵 · (𝑘 + 1))) = ((𝐴↑𝑐𝐵)↑(𝑘 + 1))))) |
| 105 | 5, 10, 15, 20, 28, 104 | nn0ind 12713 |
. . 3
⊢ (𝐶 ∈ ℕ0
→ ((𝐴 ∈ ℂ
∧ 𝐵 ∈ ℂ)
→ (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))) |
| 106 | 105 | com12 32 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 ∈ ℕ0
→ (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶))) |
| 107 | 106 | 3impia 1118 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0)
→ (𝐴↑𝑐(𝐵 · 𝐶)) = ((𝐴↑𝑐𝐵)↑𝐶)) |