Proof of Theorem dffltz
| Step | Hyp | Ref
| Expression |
|---|
| 1 | | simp-4r 782 |
. . . . . . . . . . 11
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → 𝑎 ∈ (ℤ ∖
{0})) |
| 2 | | eldifi 4103 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ 𝑎 ∈
ℤ) |
| 3 | | eldifsni 4722 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ 𝑎 ≠
0) |
| 4 | 2, 3 | jca 514 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ (𝑎 ∈ ℤ
∧ 𝑎 ≠
0)) |
| 5 | | nnabscl 14685 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈
ℕ) |
| 6 | 1, 4, 5 | 3syl 18 |
. . . . . . . . . 10
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → (abs‘𝑎) ∈ ℕ) |
| 7 | | simp-6r 786 |
. . . . . . . . . . . . . 14
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑎 ∈ (ℤ ∖
{0})) |
| 8 | 7 | eldifad 3948 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑎 ∈ ℤ) |
| 9 | | simplr 767 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < 𝑎) |
| 10 | | elnnz 11992 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ℕ ↔ (𝑎 ∈ ℤ ∧ 0 <
𝑎)) |
| 11 | 8, 9, 10 | sylanbrc 585 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑎 ∈ ℕ) |
| 12 | | eldifsni 4722 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ (ℤ ∖ {0})
→ 𝑏 ≠
0) |
| 13 | 12 | ad6antlr 735 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ≠ 0) |
| 14 | | simplr 767 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ¬ 0 < 𝑏) |
| 15 | | eldifi 4103 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ (ℤ ∖ {0})
→ 𝑏 ∈
ℤ) |
| 16 | 15 | ad6antlr 735 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ ℤ) |
| 17 | 13, 14, 16 | negn0nposznnd 39217 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → -𝑏 ∈ ℕ) |
| 18 | | simp-7r 788 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ (ℤ ∖
{0})) |
| 19 | 18 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ ℤ) |
| 20 | | simpllr 774 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 0 < 𝑎) |
| 21 | 19, 20, 10 | sylanbrc 585 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ ℕ) |
| 22 | 17, 21 | ifclda 4501 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → if(0 < 𝑐, -𝑏, 𝑎) ∈ ℕ) |
| 23 | 11, 22 | ifclda 4501 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) → if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)) ∈ ℕ) |
| 24 | 3 | ad7antlr 737 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ≠ 0) |
| 25 | | simpllr 774 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ¬ 0 < 𝑎) |
| 26 | 2 | ad7antlr 737 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ ℤ) |
| 27 | 24, 25, 26 | negn0nposznnd 39217 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → -𝑎 ∈ ℕ) |
| 28 | | simp-6r 786 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ (ℤ ∖
{0})) |
| 29 | 28 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ ℤ) |
| 30 | | simplr 767 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 0 < 𝑏) |
| 31 | | elnnz 11992 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ ℕ ↔ (𝑏 ∈ ℤ ∧ 0 <
𝑏)) |
| 32 | 29, 30, 31 | sylanbrc 585 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ ℕ) |
| 33 | 27, 32 | ifclda 4501 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) → if(0 < 𝑐, -𝑎, 𝑏) ∈ ℕ) |
| 34 | 3 | ad6antlr 735 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ≠ 0) |
| 35 | | simplr 767 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < 𝑎) |
| 36 | 2 | ad6antlr 735 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ∈ ℤ) |
| 37 | 34, 35, 36 | negn0nposznnd 39217 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → -𝑎 ∈ ℕ) |
| 38 | 33, 37 | ifclda 4501 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) → if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎) ∈ ℕ) |
| 39 | 23, 38 | ifclda 4501 |
. . . . . . . . . 10
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)) ∈ ℕ) |
| 40 | 6, 39 | ifcld 4512 |
. . . . . . . . 9
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) ∈ ℕ) |
| 41 | | simpllr 774 |
. . . . . . . . . . 11
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → 𝑏 ∈ (ℤ ∖
{0})) |
| 42 | 15, 12 | jca 514 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ (ℤ ∖ {0})
→ (𝑏 ∈ ℤ
∧ 𝑏 ≠
0)) |
| 43 | | nnabscl 14685 |
. . . . . . . . . . 11
⊢ ((𝑏 ∈ ℤ ∧ 𝑏 ≠ 0) → (abs‘𝑏) ∈
ℕ) |
| 44 | 41, 42, 43 | 3syl 18 |
. . . . . . . . . 10
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → (abs‘𝑏) ∈ ℕ) |
| 45 | | simp-5r 784 |
. . . . . . . . . . . . . 14
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑏 ∈ (ℤ ∖
{0})) |
| 46 | 45 | eldifad 3948 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑏 ∈ ℤ) |
| 47 | | simpr 487 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < 𝑏) |
| 48 | 46, 47, 31 | sylanbrc 585 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑏 ∈ ℕ) |
| 49 | | simp-5r 784 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ (ℤ ∖
{0})) |
| 50 | 49 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ ℤ) |
| 51 | | simpr 487 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 0 < 𝑐) |
| 52 | | elnnz 11992 |
. . . . . . . . . . . . . 14
⊢ (𝑐 ∈ ℕ ↔ (𝑐 ∈ ℤ ∧ 0 <
𝑐)) |
| 53 | 50, 51, 52 | sylanbrc 585 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ ℕ) |
| 54 | | eldifsni 4722 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ (ℤ ∖ {0})
→ 𝑐 ≠
0) |
| 55 | 54 | ad5antlr 733 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ≠ 0) |
| 56 | | simpr 487 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ 0 < 𝑐) |
| 57 | | eldifi 4103 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ (ℤ ∖ {0})
→ 𝑐 ∈
ℤ) |
| 58 | 57 | ad5antlr 733 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ∈ ℤ) |
| 59 | 55, 56, 58 | negn0nposznnd 39217 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -𝑐 ∈ ℕ) |
| 60 | 53, 59 | ifclda 4501 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → if(0 < 𝑐, 𝑐, -𝑐) ∈ ℕ) |
| 61 | 48, 60 | ifclda 4501 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 0 < 𝑎) → if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)) ∈ ℕ) |
| 62 | | simp-5r 784 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ (ℤ ∖
{0})) |
| 63 | 62 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ ℤ) |
| 64 | | simpr 487 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 0 < 𝑐) |
| 65 | 63, 64, 52 | sylanbrc 585 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ ℕ) |
| 66 | 54 | ad5antlr 733 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ≠ 0) |
| 67 | | simpr 487 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ 0 < 𝑐) |
| 68 | 57 | ad5antlr 733 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ∈ ℤ) |
| 69 | 66, 67, 68 | negn0nposznnd 39217 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -𝑐 ∈ ℕ) |
| 70 | 65, 69 | ifclda 4501 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ 0 < 𝑏) → if(0 < 𝑐, 𝑐, -𝑐) ∈ ℕ) |
| 71 | 12 | ad5antlr 733 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ≠ 0) |
| 72 | | simpr 487 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < 𝑏) |
| 73 | 15 | ad5antlr 733 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ∈ ℤ) |
| 74 | 71, 72, 73 | negn0nposznnd 39217 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → -𝑏 ∈ ℕ) |
| 75 | 70, 74 | ifclda 4501 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ 0 < 𝑎) → if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏) ∈ ℕ) |
| 76 | 61, 75 | ifclda 4501 |
. . . . . . . . . 10
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)) ∈ ℕ) |
| 77 | 44, 76 | ifcld 4512 |
. . . . . . . . 9
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) ∈ ℕ) |
| 78 | | simpllr 774 |
. . . . . . . . . . . 12
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑐 ∈ (ℤ ∖
{0})) |
| 79 | 78 | eldifad 3948 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑐 ∈
ℤ) |
| 80 | 78, 54 | syl 17 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑐 ≠ 0) |
| 81 | | nnabscl 14685 |
. . . . . . . . . . 11
⊢ ((𝑐 ∈ ℤ ∧ 𝑐 ≠ 0) → (abs‘𝑐) ∈
ℕ) |
| 82 | 79, 80, 81 | syl2anc 586 |
. . . . . . . . . 10
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑐) ∈
ℕ) |
| 83 | | simp-5r 784 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑐 ∈ (ℤ ∖
{0})) |
| 84 | 83 | eldifad 3948 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑐 ∈ ℤ) |
| 85 | | simp-7r 788 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑎 ∈ (ℤ ∖
{0})) |
| 86 | 85 | eldifad 3948 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑎 ∈ ℤ) |
| 87 | 86 | zred 12088 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑎 ∈ ℝ) |
| 88 | | eluzge3nn 12291 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈
(ℤ≥‘3) → 𝑛 ∈ ℕ) |
| 89 | 88 | ad7antr 736 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑛 ∈ ℕ) |
| 90 | 89 | nnnn0d 11956 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑛 ∈ ℕ0) |
| 91 | 87, 90 | reexpcld 13528 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → (𝑎↑𝑛) ∈ ℝ) |
| 92 | | simp-6r 786 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑏 ∈ (ℤ ∖
{0})) |
| 93 | 92 | eldifad 3948 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑏 ∈ ℤ) |
| 94 | 93 | zred 12088 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑏 ∈ ℝ) |
| 95 | 94, 90 | reexpcld 13528 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → (𝑏↑𝑛) ∈ ℝ) |
| 96 | | simplr 767 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < 𝑎) |
| 97 | | simpllr 774 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → ¬ (𝑛 / 2) ∈
ℕ) |
| 98 | 87, 89, 97 | oexpreposd 39228 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → (0 < 𝑎 ↔ 0 < (𝑎↑𝑛))) |
| 99 | 96, 98 | mpbid 234 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < (𝑎↑𝑛)) |
| 100 | | simpr 487 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < 𝑏) |
| 101 | 94, 89, 97 | oexpreposd 39228 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → (0 < 𝑏 ↔ 0 < (𝑏↑𝑛))) |
| 102 | 100, 101 | mpbid 234 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < (𝑏↑𝑛)) |
| 103 | 91, 95, 99, 102 | addgt0d 11215 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 104 | | simp-4r 782 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 105 | 103, 104 | breqtrd 5092 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < (𝑐↑𝑛)) |
| 106 | 84 | zred 12088 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑐 ∈ ℝ) |
| 107 | 106, 89, 97 | oexpreposd 39228 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → (0 < 𝑐 ↔ 0 < (𝑐↑𝑛))) |
| 108 | 105, 107 | mpbird 259 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 0 < 𝑐) |
| 109 | 84, 108, 52 | sylanbrc 585 |
. . . . . . . . . . . 12
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → 𝑐 ∈ ℕ) |
| 110 | | simp-8r 790 |
. . . . . . . . . . . . . . 15
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ (ℤ ∖
{0})) |
| 111 | 110 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ ℤ) |
| 112 | | simpllr 774 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 0 < 𝑎) |
| 113 | 111, 112,
10 | sylanbrc 585 |
. . . . . . . . . . . . 13
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ ℕ) |
| 114 | | simp-7r 788 |
. . . . . . . . . . . . . . 15
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ (ℤ ∖
{0})) |
| 115 | 114, 12 | syl 17 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ≠ 0) |
| 116 | | simplr 767 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ 0 < 𝑏) |
| 117 | 114 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ ℤ) |
| 118 | 115, 116,
117 | negn0nposznnd 39217 |
. . . . . . . . . . . . 13
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -𝑏 ∈ ℕ) |
| 119 | 113, 118 | ifclda 4501 |
. . . . . . . . . . . 12
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → if(0 < 𝑐, 𝑎, -𝑏) ∈ ℕ) |
| 120 | 109, 119 | ifclda 4501 |
. . . . . . . . . . 11
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) → if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)) ∈ ℕ) |
| 121 | | simp-7r 788 |
. . . . . . . . . . . . . . 15
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ (ℤ ∖
{0})) |
| 122 | 121 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ ℤ) |
| 123 | | simplr 767 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 0 < 𝑏) |
| 124 | 122, 123,
31 | sylanbrc 585 |
. . . . . . . . . . . . 13
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ ℕ) |
| 125 | | simp-8r 790 |
. . . . . . . . . . . . . . 15
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ (ℤ ∖
{0})) |
| 126 | 125, 3 | syl 17 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ≠ 0) |
| 127 | | simpllr 774 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ 0 < 𝑎) |
| 128 | 125 | eldifad 3948 |
. . . . . . . . . . . . . 14
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ ℤ) |
| 129 | 126, 127,
128 | negn0nposznnd 39217 |
. . . . . . . . . . . . 13
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -𝑎 ∈ ℕ) |
| 130 | 124, 129 | ifclda 4501 |
. . . . . . . . . . . 12
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) → if(0 < 𝑐, 𝑏, -𝑎) ∈ ℕ) |
| 131 | | simp-5r 784 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑐 ∈ (ℤ ∖
{0})) |
| 132 | 131, 54 | syl 17 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑐 ≠ 0) |
| 133 | | simp-7r 788 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ∈ (ℤ ∖
{0})) |
| 134 | 133 | eldifad 3948 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ∈ ℤ) |
| 135 | 134 | zred 12088 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ∈ ℝ) |
| 136 | 88 | ad7antr 736 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑛 ∈ ℕ) |
| 137 | 136 | nnnn0d 11956 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑛 ∈ ℕ0) |
| 138 | 135, 137 | reexpcld 13528 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑎↑𝑛) ∈ ℝ) |
| 139 | | simp-6r 786 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ∈ (ℤ ∖
{0})) |
| 140 | 139 | eldifad 3948 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ∈ ℤ) |
| 141 | 140 | zred 12088 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ∈ ℝ) |
| 142 | 141, 137 | reexpcld 13528 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑏↑𝑛) ∈ ℝ) |
| 143 | 138, 142 | readdcld 10670 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑎↑𝑛) + (𝑏↑𝑛)) ∈ ℝ) |
| 144 | | 0red 10644 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 0 ∈
ℝ) |
| 145 | 3 | neneqd 3021 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ ¬ 𝑎 =
0) |
| 146 | 133, 145 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 𝑎 = 0) |
| 147 | | zcn 11987 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑎 ∈ ℤ → 𝑎 ∈
ℂ) |
| 148 | 133, 2, 147 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ∈ ℂ) |
| 149 | | expeq0 13460 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ((𝑎↑𝑛) = 0 ↔ 𝑎 = 0)) |
| 150 | 148, 136,
149 | syl2anc 586 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑎↑𝑛) = 0 ↔ 𝑎 = 0)) |
| 151 | 146, 150 | mtbird 327 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ (𝑎↑𝑛) = 0) |
| 152 | | simplr 767 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < 𝑎) |
| 153 | | simpllr 774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ (𝑛 / 2) ∈
ℕ) |
| 154 | 135, 136,
153 | oexpreposd 39228 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (0 < 𝑎 ↔ 0 < (𝑎↑𝑛))) |
| 155 | 152, 154 | mtbid 326 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < (𝑎↑𝑛)) |
| 156 | | ioran 980 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
((𝑎↑𝑛) = 0 ∨ 0 < (𝑎↑𝑛)) ↔ (¬ (𝑎↑𝑛) = 0 ∧ ¬ 0 < (𝑎↑𝑛))) |
| 157 | 151, 155,
156 | sylanbrc 585 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ ((𝑎↑𝑛) = 0 ∨ 0 < (𝑎↑𝑛))) |
| 158 | 138, 144 | lttrid 10778 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑎↑𝑛) < 0 ↔ ¬ ((𝑎↑𝑛) = 0 ∨ 0 < (𝑎↑𝑛)))) |
| 159 | 157, 158 | mpbird 259 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑎↑𝑛) < 0) |
| 160 | | zcn 11987 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 ∈ ℤ → 𝑏 ∈
ℂ) |
| 161 | 139, 15, 160 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ∈ ℂ) |
| 162 | 139, 12 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑏 ≠ 0) |
| 163 | | eluzelz 12254 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈
(ℤ≥‘3) → 𝑛 ∈ ℤ) |
| 164 | 163 | ad7antr 736 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑛 ∈ ℤ) |
| 165 | 161, 162,
164 | expne0d 13517 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑏↑𝑛) ≠ 0) |
| 166 | 165 | neneqd 3021 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ (𝑏↑𝑛) = 0) |
| 167 | | simpr 487 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < 𝑏) |
| 168 | 141, 136,
153 | oexpreposd 39228 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (0 < 𝑏 ↔ 0 < (𝑏↑𝑛))) |
| 169 | 167, 168 | mtbid 326 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < (𝑏↑𝑛)) |
| 170 | | ioran 980 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
((𝑏↑𝑛) = 0 ∨ 0 < (𝑏↑𝑛)) ↔ (¬ (𝑏↑𝑛) = 0 ∧ ¬ 0 < (𝑏↑𝑛))) |
| 171 | 166, 169,
170 | sylanbrc 585 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ ((𝑏↑𝑛) = 0 ∨ 0 < (𝑏↑𝑛))) |
| 172 | 142, 144 | lttrid 10778 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑏↑𝑛) < 0 ↔ ¬ ((𝑏↑𝑛) = 0 ∨ 0 < (𝑏↑𝑛)))) |
| 173 | 171, 172 | mpbird 259 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑏↑𝑛) < 0) |
| 174 | 138, 142,
144, 144, 159, 173 | lt2addd 11263 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑎↑𝑛) + (𝑏↑𝑛)) < (0 + 0)) |
| 175 | | 00id 10815 |
. . . . . . . . . . . . . . . . 17
⊢ (0 + 0) =
0 |
| 176 | 174, 175 | breqtrdi 5107 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑎↑𝑛) + (𝑏↑𝑛)) < 0) |
| 177 | 143, 144,
176 | ltnsymd 10789 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 178 | | simp-4r 782 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 179 | 178 | eqcomd 2827 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑐↑𝑛) = ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 180 | 179 | breq2d 5078 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (0 < (𝑐↑𝑛) ↔ 0 < ((𝑎↑𝑛) + (𝑏↑𝑛)))) |
| 181 | 177, 180 | mtbird 327 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < (𝑐↑𝑛)) |
| 182 | 131 | eldifad 3948 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑐 ∈ ℤ) |
| 183 | 182 | zred 12088 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑐 ∈ ℝ) |
| 184 | 183, 136,
153 | oexpreposd 39228 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (0 < 𝑐 ↔ 0 < (𝑐↑𝑛))) |
| 185 | 181, 184 | mtbird 327 |
. . . . . . . . . . . . 13
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 0 < 𝑐) |
| 186 | 132, 185,
182 | negn0nposznnd 39217 |
. . . . . . . . . . . 12
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → -𝑐 ∈ ℕ) |
| 187 | 130, 186 | ifclda 4501 |
. . . . . . . . . . 11
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) → if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐) ∈ ℕ) |
| 188 | 120, 187 | ifclda 4501 |
. . . . . . . . . 10
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) → if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)) ∈ ℕ) |
| 189 | 82, 188 | ifclda 4501 |
. . . . . . . . 9
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))) ∈ ℕ) |
| 190 | | oveq1 7163 |
. . . . . . . . . . . 12
⊢ (𝑥 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) → (𝑥↑𝑛) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛)) |
| 191 | 190 | oveq1d 7171 |
. . . . . . . . . . 11
⊢ (𝑥 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) → ((𝑥↑𝑛) + (𝑦↑𝑛)) = ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (𝑦↑𝑛))) |
| 192 | 191 | eqeq1d 2823 |
. . . . . . . . . 10
⊢ (𝑥 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) → (((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛))) |
| 193 | 192 | adantl 484 |
. . . . . . . . 9
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 𝑥 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))) → (((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛))) |
| 194 | | oveq1 7163 |
. . . . . . . . . . . 12
⊢ (𝑦 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) → (𝑦↑𝑛) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) |
| 195 | 194 | oveq2d 7172 |
. . . . . . . . . . 11
⊢ (𝑦 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) → ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (𝑦↑𝑛)) = ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛))) |
| 196 | 195 | eqeq1d 2823 |
. . . . . . . . . 10
⊢ (𝑦 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) → (((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (𝑧↑𝑛))) |
| 197 | 196 | adantl 484 |
. . . . . . . . 9
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 𝑦 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))) → (((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (𝑧↑𝑛))) |
| 198 | | oveq1 7163 |
. . . . . . . . . . 11
⊢ (𝑧 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))) → (𝑧↑𝑛) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛)) |
| 199 | 198 | eqeq2d 2832 |
. . . . . . . . . 10
⊢ (𝑧 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))) → (((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (𝑧↑𝑛) ↔ ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛))) |
| 200 | 199 | adantl 484 |
. . . . . . . . 9
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ 𝑧 = if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))) → (((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (𝑧↑𝑛) ↔ ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛))) |
| 201 | | simplr 767 |
. . . . . . . . . . . 12
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 202 | | simp-5r 784 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑎 ∈ (ℤ ∖
{0})) |
| 203 | 202 | eldifad 3948 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑎 ∈
ℤ) |
| 204 | 203 | zred 12088 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑎 ∈
ℝ) |
| 205 | | absresq 14662 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ ℝ →
((abs‘𝑎)↑2) =
(𝑎↑2)) |
| 206 | 204, 205 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑎)↑2) =
(𝑎↑2)) |
| 207 | 206 | oveq1d 7171 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
(((abs‘𝑎)↑2)↑(𝑛 / 2)) = ((𝑎↑2)↑(𝑛 / 2))) |
| 208 | 202, 2, 147 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑎 ∈
ℂ) |
| 209 | 208 | abscld 14796 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑎) ∈
ℝ) |
| 210 | 209 | recnd 10669 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑎) ∈
ℂ) |
| 211 | | simpr 487 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑛 / 2) ∈
ℕ) |
| 212 | 211 | nnnn0d 11956 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑛 / 2) ∈
ℕ0) |
| 213 | | 2nn0 11915 |
. . . . . . . . . . . . . . . . 17
⊢ 2 ∈
ℕ0 |
| 214 | 213 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 2 ∈
ℕ0) |
| 215 | 210, 212,
214 | expmuld 13514 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑎)↑(2
· (𝑛 / 2))) =
(((abs‘𝑎)↑2)↑(𝑛 / 2))) |
| 216 | 208, 212,
214 | expmuld 13514 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑎↑(2 · (𝑛 / 2))) = ((𝑎↑2)↑(𝑛 / 2))) |
| 217 | 207, 215,
216 | 3eqtr4d 2866 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑎)↑(2
· (𝑛 / 2))) = (𝑎↑(2 · (𝑛 / 2)))) |
| 218 | | simp-5l 783 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑛 ∈
(ℤ≥‘3)) |
| 219 | | nncn 11646 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
| 220 | 218, 88, 219 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑛 ∈
ℂ) |
| 221 | | 2cnd 11716 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 2 ∈
ℂ) |
| 222 | | 2ne0 11742 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ≠
0 |
| 223 | 222 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 2 ≠
0) |
| 224 | 220, 221,
223 | divcan2d 11418 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (2 ·
(𝑛 / 2)) = 𝑛) |
| 225 | 224 | eqcomd 2827 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑛 = (2 · (𝑛 / 2))) |
| 226 | 225 | oveq2d 7172 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑎)↑𝑛) = ((abs‘𝑎)↑(2 · (𝑛 / 2)))) |
| 227 | 225 | oveq2d 7172 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑎↑𝑛) = (𝑎↑(2 · (𝑛 / 2)))) |
| 228 | 217, 226,
227 | 3eqtr4d 2866 |
. . . . . . . . . . . . 13
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑎)↑𝑛) = (𝑎↑𝑛)) |
| 229 | | simp-4r 782 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑏 ∈ (ℤ ∖
{0})) |
| 230 | 229 | eldifad 3948 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑏 ∈
ℤ) |
| 231 | 230 | zred 12088 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑏 ∈
ℝ) |
| 232 | | absresq 14662 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ∈ ℝ →
((abs‘𝑏)↑2) =
(𝑏↑2)) |
| 233 | 231, 232 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑏)↑2) =
(𝑏↑2)) |
| 234 | 233 | oveq1d 7171 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
(((abs‘𝑏)↑2)↑(𝑛 / 2)) = ((𝑏↑2)↑(𝑛 / 2))) |
| 235 | 229, 15, 160 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑏 ∈
ℂ) |
| 236 | 235 | abscld 14796 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑏) ∈
ℝ) |
| 237 | 236 | recnd 10669 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑏) ∈
ℂ) |
| 238 | 237, 212,
214 | expmuld 13514 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑏)↑(2
· (𝑛 / 2))) =
(((abs‘𝑏)↑2)↑(𝑛 / 2))) |
| 239 | 235, 212,
214 | expmuld 13514 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑏↑(2 · (𝑛 / 2))) = ((𝑏↑2)↑(𝑛 / 2))) |
| 240 | 234, 238,
239 | 3eqtr4d 2866 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑏)↑(2
· (𝑛 / 2))) = (𝑏↑(2 · (𝑛 / 2)))) |
| 241 | 225 | oveq2d 7172 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑏)↑𝑛) = ((abs‘𝑏)↑(2 · (𝑛 / 2)))) |
| 242 | 225 | oveq2d 7172 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑏↑𝑛) = (𝑏↑(2 · (𝑛 / 2)))) |
| 243 | 240, 241,
242 | 3eqtr4d 2866 |
. . . . . . . . . . . . 13
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑏)↑𝑛) = (𝑏↑𝑛)) |
| 244 | 228, 243 | oveq12d 7174 |
. . . . . . . . . . . 12
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
(((abs‘𝑎)↑𝑛) + ((abs‘𝑏)↑𝑛)) = ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 245 | 79 | zred 12088 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑐 ∈
ℝ) |
| 246 | | absresq 14662 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 ∈ ℝ →
((abs‘𝑐)↑2) =
(𝑐↑2)) |
| 247 | 245, 246 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑐)↑2) =
(𝑐↑2)) |
| 248 | 247 | oveq1d 7171 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
(((abs‘𝑐)↑2)↑(𝑛 / 2)) = ((𝑐↑2)↑(𝑛 / 2))) |
| 249 | | zcn 11987 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 ∈ ℤ → 𝑐 ∈
ℂ) |
| 250 | 78, 57, 249 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → 𝑐 ∈
ℂ) |
| 251 | 250 | abscld 14796 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑐) ∈
ℝ) |
| 252 | 251 | recnd 10669 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (abs‘𝑐) ∈
ℂ) |
| 253 | 252, 212,
214 | expmuld 13514 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑐)↑(2
· (𝑛 / 2))) =
(((abs‘𝑐)↑2)↑(𝑛 / 2))) |
| 254 | 250, 212,
214 | expmuld 13514 |
. . . . . . . . . . . . . 14
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑐↑(2 · (𝑛 / 2))) = ((𝑐↑2)↑(𝑛 / 2))) |
| 255 | 248, 253,
254 | 3eqtr4d 2866 |
. . . . . . . . . . . . 13
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑐)↑(2
· (𝑛 / 2))) = (𝑐↑(2 · (𝑛 / 2)))) |
| 256 | 225 | oveq2d 7172 |
. . . . . . . . . . . . 13
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑐)↑𝑛) = ((abs‘𝑐)↑(2 · (𝑛 / 2)))) |
| 257 | 225 | oveq2d 7172 |
. . . . . . . . . . . . 13
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (𝑐↑𝑛) = (𝑐↑(2 · (𝑛 / 2)))) |
| 258 | 255, 256,
257 | 3eqtr4d 2866 |
. . . . . . . . . . . 12
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
((abs‘𝑐)↑𝑛) = (𝑐↑𝑛)) |
| 259 | 201, 244,
258 | 3eqtr4d 2866 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) →
(((abs‘𝑎)↑𝑛) + ((abs‘𝑏)↑𝑛)) = ((abs‘𝑐)↑𝑛)) |
| 260 | | iftrue 4473 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 / 2) ∈ ℕ →
if((𝑛 / 2) ∈ ℕ,
(abs‘𝑎), if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) = (abs‘𝑎)) |
| 261 | 260 | oveq1d 7171 |
. . . . . . . . . . . . 13
⊢ ((𝑛 / 2) ∈ ℕ →
(if((𝑛 / 2) ∈ ℕ,
(abs‘𝑎), if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) = ((abs‘𝑎)↑𝑛)) |
| 262 | | iftrue 4473 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 / 2) ∈ ℕ →
if((𝑛 / 2) ∈ ℕ,
(abs‘𝑏), if(0 <
𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) = (abs‘𝑏)) |
| 263 | 262 | oveq1d 7171 |
. . . . . . . . . . . . 13
⊢ ((𝑛 / 2) ∈ ℕ →
(if((𝑛 / 2) ∈ ℕ,
(abs‘𝑏), if(0 <
𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛) = ((abs‘𝑏)↑𝑛)) |
| 264 | 261, 263 | oveq12d 7174 |
. . . . . . . . . . . 12
⊢ ((𝑛 / 2) ∈ ℕ →
((if((𝑛 / 2) ∈
ℕ, (abs‘𝑎),
if(0 < 𝑎, if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (((abs‘𝑎)↑𝑛) + ((abs‘𝑏)↑𝑛))) |
| 265 | 264 | adantl 484 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → ((if((𝑛 / 2) ∈ ℕ,
(abs‘𝑎), if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (((abs‘𝑎)↑𝑛) + ((abs‘𝑏)↑𝑛))) |
| 266 | | iftrue 4473 |
. . . . . . . . . . . . 13
⊢ ((𝑛 / 2) ∈ ℕ →
if((𝑛 / 2) ∈ ℕ,
(abs‘𝑐), if(0 <
𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))) = (abs‘𝑐)) |
| 267 | 266 | oveq1d 7171 |
. . . . . . . . . . . 12
⊢ ((𝑛 / 2) ∈ ℕ →
(if((𝑛 / 2) ∈ ℕ,
(abs‘𝑐), if(0 <
𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛) = ((abs‘𝑐)↑𝑛)) |
| 268 | 267 | adantl 484 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → (if((𝑛 / 2) ∈ ℕ,
(abs‘𝑐), if(0 <
𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛) = ((abs‘𝑐)↑𝑛)) |
| 269 | 259, 265,
268 | 3eqtr4d 2866 |
. . . . . . . . . 10
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ (𝑛 / 2) ∈ ℕ) → ((if((𝑛 / 2) ∈ ℕ,
(abs‘𝑎), if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛)) |
| 270 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)) = 𝑎) |
| 271 | 270 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) = (𝑎↑𝑛)) |
| 272 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)) = 𝑏) |
| 273 | 272 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛) = (𝑏↑𝑛)) |
| 274 | 271, 273 | oveq12d 7174 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑏 → ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 275 | 274 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 276 | | iftrue 4473 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)) = 𝑐) |
| 277 | 276 | oveq1d 7171 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑏 → (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛) = (𝑐↑𝑛)) |
| 278 | 277 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛) = (𝑐↑𝑛)) |
| 279 | 104, 275,
278 | 3eqtr4d 2866 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ 0 < 𝑏) → ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛)) |
| 280 | | simp-7r 788 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ (ℤ ∖
{0})) |
| 281 | 280, 15, 160 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ ℂ) |
| 282 | | simp-8l 789 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑛 ∈
(ℤ≥‘3)) |
| 283 | 282, 88 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑛 ∈ ℕ) |
| 284 | | simp-4r 782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ¬ (𝑛 / 2) ∈
ℕ) |
| 285 | | 2nn 11711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℕ |
| 286 | | nndivdvds 15616 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ ℕ ∧ 2 ∈
ℕ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
| 287 | 283, 285,
286 | sylancl 588 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
| 288 | 284, 287 | mtbird 327 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ¬ 2 ∥ 𝑛) |
| 289 | | oexpneg 15694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑏 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2
∥ 𝑛) → (-𝑏↑𝑛) = -(𝑏↑𝑛)) |
| 290 | 281, 283,
288, 289 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (-𝑏↑𝑛) = -(𝑏↑𝑛)) |
| 291 | 290 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((-𝑏↑𝑛) + (𝑐↑𝑛)) = (-(𝑏↑𝑛) + (𝑐↑𝑛))) |
| 292 | | nnnn0 11905 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
| 293 | 282, 88, 292 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑛 ∈ ℕ0) |
| 294 | 281, 293 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑏↑𝑛) ∈ ℂ) |
| 295 | 294 | negcld 10984 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → -(𝑏↑𝑛) ∈ ℂ) |
| 296 | | simp-6r 786 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ (ℤ ∖
{0})) |
| 297 | 296, 57, 249 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ ℂ) |
| 298 | 297, 293 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑐↑𝑛) ∈ ℂ) |
| 299 | 295, 298 | addcomd 10842 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (-(𝑏↑𝑛) + (𝑐↑𝑛)) = ((𝑐↑𝑛) + -(𝑏↑𝑛))) |
| 300 | 298, 294 | negsubd 11003 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑐↑𝑛) + -(𝑏↑𝑛)) = ((𝑐↑𝑛) − (𝑏↑𝑛))) |
| 301 | 299, 300 | eqtrd 2856 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (-(𝑏↑𝑛) + (𝑐↑𝑛)) = ((𝑐↑𝑛) − (𝑏↑𝑛))) |
| 302 | 110, 2, 147 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ ℂ) |
| 303 | 302, 293 | expcld 13511 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑎↑𝑛) ∈ ℂ) |
| 304 | | simp-5r 784 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 305 | 304 | eqcomd 2827 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑐↑𝑛) = ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 306 | 303, 294,
305 | mvrraddd 11052 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑐↑𝑛) − (𝑏↑𝑛)) = (𝑎↑𝑛)) |
| 307 | 291, 301,
306 | 3eqtrd 2860 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((-𝑏↑𝑛) + (𝑐↑𝑛)) = (𝑎↑𝑛)) |
| 308 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 <
𝑐 → if(0 < 𝑐, -𝑏, 𝑎) = -𝑏) |
| 309 | 308 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → (if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) = (-𝑏↑𝑛)) |
| 310 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 <
𝑐 → if(0 < 𝑐, 𝑐, -𝑐) = 𝑐) |
| 311 | 310 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛) = (𝑐↑𝑛)) |
| 312 | 309, 311 | oveq12d 7174 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((-𝑏↑𝑛) + (𝑐↑𝑛))) |
| 313 | 312 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((-𝑏↑𝑛) + (𝑐↑𝑛))) |
| 314 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → if(0 < 𝑐, 𝑎, -𝑏) = 𝑎) |
| 315 | 314 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛) = (𝑎↑𝑛)) |
| 316 | 315 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛) = (𝑎↑𝑛)) |
| 317 | 307, 313,
316 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ 0 < 𝑐) → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛)) |
| 318 | | simp-8r 790 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ (ℤ ∖
{0})) |
| 319 | 318, 2, 147 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ ℂ) |
| 320 | 88 | ad8antr 738 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑛 ∈ ℕ) |
| 321 | 320 | nnnn0d 11956 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑛 ∈ ℕ0) |
| 322 | 319, 321 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑎↑𝑛) ∈ ℂ) |
| 323 | | simp-6r 786 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ∈ (ℤ ∖
{0})) |
| 324 | 323, 57, 249 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ∈ ℂ) |
| 325 | 324, 321 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑐↑𝑛) ∈ ℂ) |
| 326 | 322, 325 | negsubd 11003 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) + -(𝑐↑𝑛)) = ((𝑎↑𝑛) − (𝑐↑𝑛))) |
| 327 | 322, 325 | subcld 10997 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) − (𝑐↑𝑛)) ∈ ℂ) |
| 328 | 114, 15, 160 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ ℂ) |
| 329 | 328, 321 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑏↑𝑛) ∈ ℂ) |
| 330 | 329 | negcld 10984 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -(𝑏↑𝑛) ∈ ℂ) |
| 331 | | simp-5r 784 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 332 | 322, 329,
331 | mvlraddd 11050 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑎↑𝑛) = ((𝑐↑𝑛) − (𝑏↑𝑛))) |
| 333 | 325, 322 | pncan3d 11000 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑐↑𝑛) + ((𝑎↑𝑛) − (𝑐↑𝑛))) = (𝑎↑𝑛)) |
| 334 | 325, 329 | negsubd 11003 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑐↑𝑛) + -(𝑏↑𝑛)) = ((𝑐↑𝑛) − (𝑏↑𝑛))) |
| 335 | 332, 333,
334 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑐↑𝑛) + ((𝑎↑𝑛) − (𝑐↑𝑛))) = ((𝑐↑𝑛) + -(𝑏↑𝑛))) |
| 336 | 325, 327,
330, 335 | addcanad 10845 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) − (𝑐↑𝑛)) = -(𝑏↑𝑛)) |
| 337 | 326, 336 | eqtrd 2856 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) + -(𝑐↑𝑛)) = -(𝑏↑𝑛)) |
| 338 | | simp-4r 782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ (𝑛 / 2) ∈
ℕ) |
| 339 | 320, 285,
286 | sylancl 588 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
| 340 | 338, 339 | mtbird 327 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ 2 ∥ 𝑛) |
| 341 | | oexpneg 15694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2
∥ 𝑛) → (-𝑐↑𝑛) = -(𝑐↑𝑛)) |
| 342 | 324, 320,
340, 341 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (-𝑐↑𝑛) = -(𝑐↑𝑛)) |
| 343 | 342 | oveq2d 7172 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) + (-𝑐↑𝑛)) = ((𝑎↑𝑛) + -(𝑐↑𝑛))) |
| 344 | 328, 320,
340, 289 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (-𝑏↑𝑛) = -(𝑏↑𝑛)) |
| 345 | 337, 343,
344 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) + (-𝑐↑𝑛)) = (-𝑏↑𝑛)) |
| 346 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, -𝑏, 𝑎) = 𝑎) |
| 347 | 346 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, -𝑏, 𝑎)↑𝑛) = (𝑎↑𝑛)) |
| 348 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, 𝑐, -𝑐) = -𝑐) |
| 349 | 348 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, 𝑐, -𝑐)↑𝑛) = (-𝑐↑𝑛)) |
| 350 | 347, 349 | oveq12d 7174 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → ((if(0 <
𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((𝑎↑𝑛) + (-𝑐↑𝑛))) |
| 351 | 350 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((𝑎↑𝑛) + (-𝑐↑𝑛))) |
| 352 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, 𝑎, -𝑏) = -𝑏) |
| 353 | 352 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, 𝑎, -𝑏)↑𝑛) = (-𝑏↑𝑛)) |
| 354 | 353 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛) = (-𝑏↑𝑛)) |
| 355 | 345, 351,
354 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛)) |
| 356 | 317, 355 | pm2.61dan 811 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛)) |
| 357 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)) = if(0 < 𝑐, -𝑏, 𝑎)) |
| 358 | 357 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) = (if(0 < 𝑐, -𝑏, 𝑎)↑𝑛)) |
| 359 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)) = if(0 < 𝑐, 𝑐, -𝑐)) |
| 360 | 359 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛) = (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) |
| 361 | 358, 360 | oveq12d 7174 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑏 → ((if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛))) |
| 362 | 361 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛))) |
| 363 | | iffalse 4476 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)) = if(0 < 𝑐, 𝑎, -𝑏)) |
| 364 | 363 | oveq1d 7171 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛) = (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛)) |
| 365 | 364 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛) = (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛)) |
| 366 | 356, 362,
365 | 3eqtr4d 2866 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) ∧ ¬ 0 < 𝑏) → ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛)) |
| 367 | 279, 366 | pm2.61dan 811 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) → ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) = (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛)) |
| 368 | | iftrue 4473 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑎 → if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)) = if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))) |
| 369 | 368 | oveq1d 7171 |
. . . . . . . . . . . . . . 15
⊢ (0 <
𝑎 → (if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) = (if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛)) |
| 370 | | iftrue 4473 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑎 → if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)) = if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))) |
| 371 | 370 | oveq1d 7171 |
. . . . . . . . . . . . . . 15
⊢ (0 <
𝑎 → (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛) = (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛)) |
| 372 | 369, 371 | oveq12d 7174 |
. . . . . . . . . . . . . 14
⊢ (0 <
𝑎 → ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛))) |
| 373 | 372 | adantl 484 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) → ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = ((if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) + (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛))) |
| 374 | | iftrue 4473 |
. . . . . . . . . . . . . . 15
⊢ (0 <
𝑎 → if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)) = if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))) |
| 375 | 374 | oveq1d 7171 |
. . . . . . . . . . . . . 14
⊢ (0 <
𝑎 → (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛) = (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛)) |
| 376 | 375 | adantl 484 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) → (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛) = (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛)) |
| 377 | 367, 373,
376 | 3eqtr4d 2866 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ 0 < 𝑎) → ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛)) |
| 378 | | simp-8r 790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ (ℤ ∖
{0})) |
| 379 | 378, 2, 147 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑎 ∈ ℂ) |
| 380 | 88 | ad8antr 738 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑛 ∈ ℕ) |
| 381 | | simp-4r 782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ¬ (𝑛 / 2) ∈
ℕ) |
| 382 | 380, 285,
286 | sylancl 588 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
| 383 | 381, 382 | mtbird 327 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ¬ 2 ∥ 𝑛) |
| 384 | | oexpneg 15694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2
∥ 𝑛) → (-𝑎↑𝑛) = -(𝑎↑𝑛)) |
| 385 | 379, 380,
383, 384 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (-𝑎↑𝑛) = -(𝑎↑𝑛)) |
| 386 | 385 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((-𝑎↑𝑛) + (𝑐↑𝑛)) = (-(𝑎↑𝑛) + (𝑐↑𝑛))) |
| 387 | 380 | nnnn0d 11956 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑛 ∈ ℕ0) |
| 388 | 379, 387 | expcld 13511 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑎↑𝑛) ∈ ℂ) |
| 389 | 388 | negcld 10984 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → -(𝑎↑𝑛) ∈ ℂ) |
| 390 | | simp-6r 786 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ (ℤ ∖
{0})) |
| 391 | 390, 57, 249 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑐 ∈ ℂ) |
| 392 | 391, 387 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑐↑𝑛) ∈ ℂ) |
| 393 | 389, 392 | addcld 10660 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (-(𝑎↑𝑛) + (𝑐↑𝑛)) ∈ ℂ) |
| 394 | 121, 15, 160 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → 𝑏 ∈ ℂ) |
| 395 | 394, 387 | expcld 13511 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (𝑏↑𝑛) ∈ ℂ) |
| 396 | 388 | negidd 10987 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑎↑𝑛) + -(𝑎↑𝑛)) = 0) |
| 397 | 396 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (((𝑎↑𝑛) + -(𝑎↑𝑛)) + (𝑐↑𝑛)) = (0 + (𝑐↑𝑛))) |
| 398 | 388, 389,
392 | addassd 10663 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (((𝑎↑𝑛) + -(𝑎↑𝑛)) + (𝑐↑𝑛)) = ((𝑎↑𝑛) + (-(𝑎↑𝑛) + (𝑐↑𝑛)))) |
| 399 | 392 | addid2d 10841 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (0 + (𝑐↑𝑛)) = (𝑐↑𝑛)) |
| 400 | 397, 398,
399 | 3eqtr3d 2864 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑎↑𝑛) + (-(𝑎↑𝑛) + (𝑐↑𝑛))) = (𝑐↑𝑛)) |
| 401 | | simp-5r 784 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 402 | 400, 401 | eqtr4d 2859 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((𝑎↑𝑛) + (-(𝑎↑𝑛) + (𝑐↑𝑛))) = ((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 403 | 388, 393,
395, 402 | addcanad 10845 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (-(𝑎↑𝑛) + (𝑐↑𝑛)) = (𝑏↑𝑛)) |
| 404 | 386, 403 | eqtrd 2856 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((-𝑎↑𝑛) + (𝑐↑𝑛)) = (𝑏↑𝑛)) |
| 405 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 <
𝑐 → if(0 < 𝑐, -𝑎, 𝑏) = -𝑎) |
| 406 | 405 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → (if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) = (-𝑎↑𝑛)) |
| 407 | 406, 311 | oveq12d 7174 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((-𝑎↑𝑛) + (𝑐↑𝑛))) |
| 408 | 407 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((-𝑎↑𝑛) + (𝑐↑𝑛))) |
| 409 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → if(0 < 𝑐, 𝑏, -𝑎) = 𝑏) |
| 410 | 409 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛) = (𝑏↑𝑛)) |
| 411 | 410 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛) = (𝑏↑𝑛)) |
| 412 | 404, 408,
411 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ 0 < 𝑐) → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛)) |
| 413 | | simp-7r 788 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ (ℤ ∖
{0})) |
| 414 | 413, 15, 160 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑏 ∈ ℂ) |
| 415 | | simp-8l 789 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑛 ∈
(ℤ≥‘3)) |
| 416 | 415, 88, 292 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑛 ∈ ℕ0) |
| 417 | 414, 416 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑏↑𝑛) ∈ ℂ) |
| 418 | 417 | negcld 10984 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -(𝑏↑𝑛) ∈ ℂ) |
| 419 | | simp-6r 786 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ∈ (ℤ ∖
{0})) |
| 420 | 419, 57, 249 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑐 ∈ ℂ) |
| 421 | 420, 416 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑐↑𝑛) ∈ ℂ) |
| 422 | 418, 421 | addcomd 10842 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (-(𝑏↑𝑛) + (𝑐↑𝑛)) = ((𝑐↑𝑛) + -(𝑏↑𝑛))) |
| 423 | 421, 417 | negsubd 11003 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑐↑𝑛) + -(𝑏↑𝑛)) = ((𝑐↑𝑛) − (𝑏↑𝑛))) |
| 424 | | simp-5r 784 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 425 | 424 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (((𝑎↑𝑛) + (𝑏↑𝑛)) − (𝑏↑𝑛)) = ((𝑐↑𝑛) − (𝑏↑𝑛))) |
| 426 | 125, 2, 147 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑎 ∈ ℂ) |
| 427 | 426, 416 | expcld 13511 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑎↑𝑛) ∈ ℂ) |
| 428 | 427, 417 | pncand 10998 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (((𝑎↑𝑛) + (𝑏↑𝑛)) − (𝑏↑𝑛)) = (𝑎↑𝑛)) |
| 429 | 425, 428 | eqtr3d 2858 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑐↑𝑛) − (𝑏↑𝑛)) = (𝑎↑𝑛)) |
| 430 | 422, 423,
429 | 3eqtrd 2860 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (-(𝑏↑𝑛) + (𝑐↑𝑛)) = (𝑎↑𝑛)) |
| 431 | 430 | negeqd 10880 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -(-(𝑏↑𝑛) + (𝑐↑𝑛)) = -(𝑎↑𝑛)) |
| 432 | 417 | negnegd 10988 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → --(𝑏↑𝑛) = (𝑏↑𝑛)) |
| 433 | 432 | eqcomd 2827 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (𝑏↑𝑛) = --(𝑏↑𝑛)) |
| 434 | 433 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑏↑𝑛) + -(𝑐↑𝑛)) = (--(𝑏↑𝑛) + -(𝑐↑𝑛))) |
| 435 | 415, 88 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → 𝑛 ∈ ℕ) |
| 436 | | simp-4r 782 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ (𝑛 / 2) ∈
ℕ) |
| 437 | 435, 285,
286 | sylancl 588 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
| 438 | 436, 437 | mtbird 327 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ¬ 2 ∥ 𝑛) |
| 439 | 420, 435,
438, 341 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (-𝑐↑𝑛) = -(𝑐↑𝑛)) |
| 440 | 439 | oveq2d 7172 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑏↑𝑛) + (-𝑐↑𝑛)) = ((𝑏↑𝑛) + -(𝑐↑𝑛))) |
| 441 | 418, 421 | negdid 11010 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → -(-(𝑏↑𝑛) + (𝑐↑𝑛)) = (--(𝑏↑𝑛) + -(𝑐↑𝑛))) |
| 442 | 434, 440,
441 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑏↑𝑛) + (-𝑐↑𝑛)) = -(-(𝑏↑𝑛) + (𝑐↑𝑛))) |
| 443 | 426, 435,
438, 384 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (-𝑎↑𝑛) = -(𝑎↑𝑛)) |
| 444 | 431, 442,
443 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((𝑏↑𝑛) + (-𝑐↑𝑛)) = (-𝑎↑𝑛)) |
| 445 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, -𝑎, 𝑏) = 𝑏) |
| 446 | 445 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, -𝑎, 𝑏)↑𝑛) = (𝑏↑𝑛)) |
| 447 | 446, 349 | oveq12d 7174 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → ((if(0 <
𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((𝑏↑𝑛) + (-𝑐↑𝑛))) |
| 448 | 447 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((𝑏↑𝑛) + (-𝑐↑𝑛))) |
| 449 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, 𝑏, -𝑎) = -𝑎) |
| 450 | 449 | oveq1d 7171 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, 𝑏, -𝑎)↑𝑛) = (-𝑎↑𝑛)) |
| 451 | 450 | adantl 484 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛) = (-𝑎↑𝑛)) |
| 452 | 444, 448,
451 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) ∧ ¬ 0 < 𝑐) → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛)) |
| 453 | 412, 452 | pm2.61dan 811 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛)) |
| 454 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎) = if(0 < 𝑐, -𝑎, 𝑏)) |
| 455 | 454 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) = (if(0 < 𝑐, -𝑎, 𝑏)↑𝑛)) |
| 456 | | iftrue 4473 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏) = if(0 < 𝑐, 𝑐, -𝑐)) |
| 457 | 456 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛) = (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) |
| 458 | 455, 457 | oveq12d 7174 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑏 → ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛))) |
| 459 | 458 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) → ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛))) |
| 460 | | iftrue 4473 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐) = if(0 < 𝑐, 𝑏, -𝑎)) |
| 461 | 460 | oveq1d 7171 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑏 → (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛) = (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛)) |
| 462 | 461 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) → (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛) = (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛)) |
| 463 | 453, 459,
462 | 3eqtr4d 2866 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ 0 < 𝑏) → ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛)) |
| 464 | 178 | negeqd 10880 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → -((𝑎↑𝑛) + (𝑏↑𝑛)) = -(𝑐↑𝑛)) |
| 465 | 136, 285,
286 | sylancl 588 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
| 466 | 153, 465 | mtbird 327 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ¬ 2 ∥ 𝑛) |
| 467 | 148, 136,
466, 384 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (-𝑎↑𝑛) = -(𝑎↑𝑛)) |
| 468 | 161, 136,
466, 289 | syl3anc 1367 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (-𝑏↑𝑛) = -(𝑏↑𝑛)) |
| 469 | 467, 468 | oveq12d 7174 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((-𝑎↑𝑛) + (-𝑏↑𝑛)) = (-(𝑎↑𝑛) + -(𝑏↑𝑛))) |
| 470 | 133, 3 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑎 ≠ 0) |
| 471 | 148, 470,
164 | expclzd 13516 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑎↑𝑛) ∈ ℂ) |
| 472 | 161, 162,
164 | expclzd 13516 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (𝑏↑𝑛) ∈ ℂ) |
| 473 | 471, 472 | negdid 11010 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → -((𝑎↑𝑛) + (𝑏↑𝑛)) = (-(𝑎↑𝑛) + -(𝑏↑𝑛))) |
| 474 | 469, 473 | eqtr4d 2859 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((-𝑎↑𝑛) + (-𝑏↑𝑛)) = -((𝑎↑𝑛) + (𝑏↑𝑛))) |
| 475 | 131, 57, 249 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → 𝑐 ∈ ℂ) |
| 476 | 475, 136,
466, 341 | syl3anc 1367 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (-𝑐↑𝑛) = -(𝑐↑𝑛)) |
| 477 | 464, 474,
476 | 3eqtr4d 2866 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((-𝑎↑𝑛) + (-𝑏↑𝑛)) = (-𝑐↑𝑛)) |
| 478 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎) = -𝑎) |
| 479 | 478 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) = (-𝑎↑𝑛)) |
| 480 | | iffalse 4476 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏) = -𝑏) |
| 481 | 480 | oveq1d 7171 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛) = (-𝑏↑𝑛)) |
| 482 | 479, 481 | oveq12d 7174 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑏 → ((if(0 <
𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = ((-𝑎↑𝑛) + (-𝑏↑𝑛))) |
| 483 | 482 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = ((-𝑎↑𝑛) + (-𝑏↑𝑛))) |
| 484 | | iffalse 4476 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐) = -𝑐) |
| 485 | 484 | oveq1d 7171 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛) = (-𝑐↑𝑛)) |
| 486 | 485 | adantl 484 |
. . . . . . . . . . . . . . 15
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛) = (-𝑐↑𝑛)) |
| 487 | 477, 483,
486 | 3eqtr4d 2866 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) ∧ ¬ 0 < 𝑏) → ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛)) |
| 488 | 463, 487 | pm2.61dan 811 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) → ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) = (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛)) |
| 489 | | iffalse 4476 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑎 → if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)) = if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)) |
| 490 | 489 | oveq1d 7171 |
. . . . . . . . . . . . . . 15
⊢ (¬ 0
< 𝑎 → (if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) = (if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛)) |
| 491 | | iffalse 4476 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑎 → if(0 <
𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)) = if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)) |
| 492 | 491 | oveq1d 7171 |
. . . . . . . . . . . . . . 15
⊢ (¬ 0
< 𝑎 → (if(0 <
𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛) = (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛)) |
| 493 | 490, 492 | oveq12d 7174 |
. . . . . . . . . . . . . 14
⊢ (¬ 0
< 𝑎 → ((if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛))) |
| 494 | 493 | adantl 484 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) → ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = ((if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) + (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛))) |
| 495 | | iffalse 4476 |
. . . . . . . . . . . . . . 15
⊢ (¬ 0
< 𝑎 → if(0 <
𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)) = if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)) |
| 496 | 495 | oveq1d 7171 |
. . . . . . . . . . . . . 14
⊢ (¬ 0
< 𝑎 → (if(0 <
𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛) = (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛)) |
| 497 | 496 | adantl 484 |
. . . . . . . . . . . . 13
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) → (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛) = (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛)) |
| 498 | 488, 494,
497 | 3eqtr4d 2866 |
. . . . . . . . . . . 12
⊢
(((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) ∧ ¬ 0 <
𝑎) → ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛)) |
| 499 | 377, 498 | pm2.61dan 811 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) = (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛)) |
| 500 | | iffalse 4476 |
. . . . . . . . . . . . . 14
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ if((𝑛 / 2) ∈
ℕ, (abs‘𝑎),
if(0 < 𝑎, if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) = if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))) |
| 501 | 500 | oveq1d 7171 |
. . . . . . . . . . . . 13
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ (if((𝑛 / 2) ∈
ℕ, (abs‘𝑎),
if(0 < 𝑎, if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) = (if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛)) |
| 502 | | iffalse 4476 |
. . . . . . . . . . . . . 14
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ if((𝑛 / 2) ∈
ℕ, (abs‘𝑏),
if(0 < 𝑎, if(0 <
𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) = if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))) |
| 503 | 502 | oveq1d 7171 |
. . . . . . . . . . . . 13
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ (if((𝑛 / 2) ∈
ℕ, (abs‘𝑏),
if(0 < 𝑎, if(0 <
𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛) = (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛)) |
| 504 | 501, 503 | oveq12d 7174 |
. . . . . . . . . . . 12
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ ((if((𝑛 / 2) ∈
ℕ, (abs‘𝑎),
if(0 < 𝑎, if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛))) |
| 505 | 504 | adantl 484 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((if((𝑛 / 2) ∈ ℕ,
(abs‘𝑎), if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = ((if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎))↑𝑛) + (if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏))↑𝑛))) |
| 506 | | iffalse 4476 |
. . . . . . . . . . . . 13
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ if((𝑛 / 2) ∈
ℕ, (abs‘𝑐),
if(0 < 𝑎, if(0 <
𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))) = if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))) |
| 507 | 506 | oveq1d 7171 |
. . . . . . . . . . . 12
⊢ (¬
(𝑛 / 2) ∈ ℕ
→ (if((𝑛 / 2) ∈
ℕ, (abs‘𝑐),
if(0 < 𝑎, if(0 <
𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛) = (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛)) |
| 508 | 507 | adantl 484 |
. . . . . . . . . . 11
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (if((𝑛 / 2) ∈ ℕ,
(abs‘𝑐), if(0 <
𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛) = (if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐))↑𝑛)) |
| 509 | 499, 505,
508 | 3eqtr4d 2866 |
. . . . . . . . . 10
⊢
((((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((if((𝑛 / 2) ∈ ℕ,
(abs‘𝑎), if(0 <
𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛)) |
| 510 | 269, 509 | pm2.61dan 811 |
. . . . . . . . 9
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → ((if((𝑛 / 2) ∈ ℕ, (abs‘𝑎), if(0 < 𝑎, if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)), if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)))↑𝑛) + (if((𝑛 / 2) ∈ ℕ, (abs‘𝑏), if(0 < 𝑎, if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)), if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)))↑𝑛)) = (if((𝑛 / 2) ∈ ℕ, (abs‘𝑐), if(0 < 𝑎, if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)), if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)))↑𝑛)) |
| 511 | 40, 77, 189, 193, 197, 200, 510 | 3rspcedvd 39154 |
. . . . . . . 8
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛)) |
| 512 | 511 | rexlimdva2 3287 |
. . . . . . 7
⊢ (((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
→ (∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛))) |
| 513 | 512 | rexlimdva 3284 |
. . . . . 6
⊢ ((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) →
(∃𝑏 ∈ (ℤ
∖ {0})∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛))) |
| 514 | 513 | rexlimdva 3284 |
. . . . 5
⊢ (𝑛 ∈
(ℤ≥‘3) → (∃𝑎 ∈ (ℤ ∖ {0})∃𝑏 ∈ (ℤ ∖
{0})∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛))) |
| 515 | 514 | reximia 3242 |
. . . 4
⊢
(∃𝑛 ∈
(ℤ≥‘3)∃𝑎 ∈ (ℤ ∖ {0})∃𝑏 ∈ (ℤ ∖
{0})∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) → ∃𝑛 ∈
(ℤ≥‘3)∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛)) |
| 516 | | nne 3020 |
. . . . . . . . . . . . 13
⊢ (¬
((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
| 517 | 516 | bicomi 226 |
. . . . . . . . . . . 12
⊢ (((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 518 | 517 | rexbii 3247 |
. . . . . . . . . . 11
⊢
(∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ∃𝑐 ∈ (ℤ ∖ {0}) ¬ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 519 | | rexnal 3238 |
. . . . . . . . . . 11
⊢
(∃𝑐 ∈
(ℤ ∖ {0}) ¬ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ¬ ∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 520 | 518, 519 | bitri 277 |
. . . . . . . . . 10
⊢
(∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 521 | 520 | rexbii 3247 |
. . . . . . . . 9
⊢
(∃𝑏 ∈
(ℤ ∖ {0})∃𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ∃𝑏 ∈ (ℤ ∖ {0}) ¬
∀𝑐 ∈ (ℤ
∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 522 | | rexnal 3238 |
. . . . . . . . 9
⊢
(∃𝑏 ∈
(ℤ ∖ {0}) ¬ ∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ¬ ∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 523 | 521, 522 | bitri 277 |
. . . . . . . 8
⊢
(∃𝑏 ∈
(ℤ ∖ {0})∃𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 524 | 523 | rexbii 3247 |
. . . . . . 7
⊢
(∃𝑎 ∈
(ℤ ∖ {0})∃𝑏 ∈ (ℤ ∖ {0})∃𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ∃𝑎 ∈ (ℤ ∖ {0}) ¬
∀𝑏 ∈ (ℤ
∖ {0})∀𝑐
∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 525 | | rexnal 3238 |
. . . . . . 7
⊢
(∃𝑎 ∈
(ℤ ∖ {0}) ¬ ∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ¬ ∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 526 | 524, 525 | bitri 277 |
. . . . . 6
⊢
(∃𝑎 ∈
(ℤ ∖ {0})∃𝑏 ∈ (ℤ ∖ {0})∃𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 527 | 526 | rexbii 3247 |
. . . . 5
⊢
(∃𝑛 ∈
(ℤ≥‘3)∃𝑎 ∈ (ℤ ∖ {0})∃𝑏 ∈ (ℤ ∖
{0})∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ∃𝑛 ∈ (ℤ≥‘3)
¬ ∀𝑎 ∈
(ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 528 | | rexnal 3238 |
. . . . 5
⊢
(∃𝑛 ∈
(ℤ≥‘3) ¬ ∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ¬ ∀𝑛 ∈
(ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 529 | 527, 528 | bitri 277 |
. . . 4
⊢
(∃𝑛 ∈
(ℤ≥‘3)∃𝑎 ∈ (ℤ ∖ {0})∃𝑏 ∈ (ℤ ∖
{0})∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ∀𝑛 ∈
(ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 530 | | nne 3020 |
. . . . . . . . . . . . 13
⊢ (¬
((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛)) |
| 531 | 530 | bicomi 226 |
. . . . . . . . . . . 12
⊢ (((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 532 | 531 | rexbii 3247 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ∃𝑧 ∈ ℕ ¬ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 533 | | rexnal 3238 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
ℕ ¬ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ¬ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 534 | 532, 533 | bitri 277 |
. . . . . . . . . 10
⊢
(∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 535 | 534 | rexbii 3247 |
. . . . . . . . 9
⊢
(∃𝑦 ∈
ℕ ∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ∃𝑦 ∈ ℕ ¬ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 536 | | rexnal 3238 |
. . . . . . . . 9
⊢
(∃𝑦 ∈
ℕ ¬ ∀𝑧
∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ¬ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 537 | 535, 536 | bitri 277 |
. . . . . . . 8
⊢
(∃𝑦 ∈
ℕ ∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 538 | 537 | rexbii 3247 |
. . . . . . 7
⊢
(∃𝑥 ∈
ℕ ∃𝑦 ∈
ℕ ∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ∃𝑥 ∈ ℕ ¬ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 539 | | rexnal 3238 |
. . . . . . 7
⊢
(∃𝑥 ∈
ℕ ¬ ∀𝑦
∈ ℕ ∀𝑧
∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ¬ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 540 | 538, 539 | bitri 277 |
. . . . . 6
⊢
(∃𝑥 ∈
ℕ ∃𝑦 ∈
ℕ ∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 541 | 540 | rexbii 3247 |
. . . . 5
⊢
(∃𝑛 ∈
(ℤ≥‘3)∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ∃𝑛 ∈ (ℤ≥‘3)
¬ ∀𝑥 ∈
ℕ ∀𝑦 ∈
ℕ ∀𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 542 | | rexnal 3238 |
. . . . 5
⊢
(∃𝑛 ∈
(ℤ≥‘3) ¬ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ¬ ∀𝑛 ∈
(ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 543 | 541, 542 | bitri 277 |
. . . 4
⊢
(∃𝑛 ∈
(ℤ≥‘3)∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ∃𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ∀𝑛 ∈
(ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 544 | 515, 529,
543 | 3imtr3i 293 |
. . 3
⊢ (¬
∀𝑛 ∈
(ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ¬ ∀𝑛 ∈
(ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 545 | 544 | con4i 114 |
. 2
⊢
(∀𝑛 ∈
(ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) → ∀𝑛 ∈
(ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 546 | | 0nnn 11674 |
. . . . . . . 8
⊢ ¬ 0
∈ ℕ |
| 547 | | disjsn 4647 |
. . . . . . . 8
⊢ ((ℕ
∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) |
| 548 | 546, 547 | mpbir 233 |
. . . . . . 7
⊢ (ℕ
∩ {0}) = ∅ |
| 549 | | disj3 4403 |
. . . . . . 7
⊢ ((ℕ
∩ {0}) = ∅ ↔ ℕ = (ℕ ∖ {0})) |
| 550 | 548, 549 | mpbi 232 |
. . . . . 6
⊢ ℕ =
(ℕ ∖ {0}) |
| 551 | | nnssz 12003 |
. . . . . . 7
⊢ ℕ
⊆ ℤ |
| 552 | | ssdif 4116 |
. . . . . . 7
⊢ (ℕ
⊆ ℤ → (ℕ ∖ {0}) ⊆ (ℤ ∖
{0})) |
| 553 | 551, 552 | ax-mp 5 |
. . . . . 6
⊢ (ℕ
∖ {0}) ⊆ (ℤ ∖ {0}) |
| 554 | 550, 553 | eqsstri 4001 |
. . . . 5
⊢ ℕ
⊆ (ℤ ∖ {0}) |
| 555 | | ssel 3961 |
. . . . . . 7
⊢ (ℕ
⊆ (ℤ ∖ {0}) → (𝑎 ∈ ℕ → 𝑎 ∈ (ℤ ∖
{0}))) |
| 556 | | ss2ralv 4035 |
. . . . . . 7
⊢ (ℕ
⊆ (ℤ ∖ {0}) → (∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))) |
| 557 | 555, 556 | imim12d 81 |
. . . . . 6
⊢ (ℕ
⊆ (ℤ ∖ {0}) → ((𝑎 ∈ (ℤ ∖ {0}) →
∀𝑏 ∈ (ℤ
∖ {0})∀𝑐
∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) → (𝑎 ∈ ℕ → ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)))) |
| 558 | 557 | ralimdv2 3176 |
. . . . 5
⊢ (ℕ
⊆ (ℤ ∖ {0}) → (∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))) |
| 559 | 554, 558 | ax-mp 5 |
. . . 4
⊢
(∀𝑎 ∈
(ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
| 560 | | oveq1 7163 |
. . . . . . 7
⊢ (𝑎 = 𝑥 → (𝑎↑𝑛) = (𝑥↑𝑛)) |
| 561 | 560 | oveq1d 7171 |
. . . . . 6
⊢ (𝑎 = 𝑥 → ((𝑎↑𝑛) + (𝑏↑𝑛)) = ((𝑥↑𝑛) + (𝑏↑𝑛))) |
| 562 | 561 | neeq1d 3075 |
. . . . 5
⊢ (𝑎 = 𝑥 → (((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑥↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))) |
| 563 | | oveq1 7163 |
. . . . . . 7
⊢ (𝑏 = 𝑦 → (𝑏↑𝑛) = (𝑦↑𝑛)) |
| 564 | 563 | oveq2d 7172 |
. . . . . 6
⊢ (𝑏 = 𝑦 → ((𝑥↑𝑛) + (𝑏↑𝑛)) = ((𝑥↑𝑛) + (𝑦↑𝑛))) |
| 565 | 564 | neeq1d 3075 |
. . . . 5
⊢ (𝑏 = 𝑦 → (((𝑥↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑐↑𝑛))) |
| 566 | | oveq1 7163 |
. . . . . 6
⊢ (𝑐 = 𝑧 → (𝑐↑𝑛) = (𝑧↑𝑛)) |
| 567 | 566 | neeq2d 3076 |
. . . . 5
⊢ (𝑐 = 𝑧 → (((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛))) |
| 568 | 562, 565,
567 | cbvral3vw 3463 |
. . . 4
⊢
(∀𝑎 ∈
ℕ ∀𝑏 ∈
ℕ ∀𝑐 ∈
ℕ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 569 | 559, 568 | sylib 220 |
. . 3
⊢
(∀𝑎 ∈
(ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖
{0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 570 | 569 | ralimi 3160 |
. 2
⊢
(∀𝑛 ∈
(ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ∀𝑛 ∈
(ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
| 571 | 545, 570 | impbii 211 |
1
⊢
(∀𝑛 ∈
(ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ∀𝑛 ∈
(ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖
{0})∀𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
