Proof of Theorem dffltz
Step | Hyp | Ref
| Expression |
1 | | simp-4r 803 |
. . . . . . . . . . 11
⊢
(((((𝑛 ∈
(ℤ≥‘3) ∧ 𝑎 ∈ (ℤ ∖ {0})) ∧ 𝑏 ∈ (ℤ ∖ {0}))
∧ 𝑐 ∈ (ℤ
∖ {0})) ∧ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) → 𝑎 ∈ (ℤ ∖
{0})) |
2 | | eldifi 3961 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ 𝑎 ∈
ℤ) |
3 | | eldifsni 4542 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ 𝑎 ≠
0) |
4 | 2, 3 | jca 507 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ (ℤ ∖ {0})
→ (𝑎 ∈ ℤ
∧ 𝑎 ≠
0)) |
5 | | nnabscl 14449 |
. . . . . . . . . . 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 811 |
. . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . 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 11721 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ ℕ ↔ (𝑎 ∈ ℤ ∧ 0 <
𝑎)) |
11 | 8, 9, 10 | sylanbrc 578 |
. . . . . . . . . . . 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 4542 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ (ℤ ∖ {0})
→ 𝑏 ≠
0) |
13 | 12 | ad6antlr 734 |
. . . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . . 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 3961 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ (ℤ ∖ {0})
→ 𝑏 ∈
ℤ) |
16 | 15 | ad6antlr 734 |
. . . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 793 |
. . . . . . . . . . . . . 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 578 |
. . . . . . . . . . . . 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 4342 |
. . . . . . . . . . . 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 4342 |
. . . . . . . . . . 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 738 |
. . . . . . . . . . . . . 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 793 |
. . . . . . . . . . . . . 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 738 |
. . . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . . 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 11721 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ ℕ ↔ (𝑏 ∈ ℤ ∧ 0 <
𝑏)) |
32 | 29, 30, 31 | sylanbrc 578 |
. . . . . . . . . . . . 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 4342 |
. . . . . . . . . . . 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 734 |
. . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . 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 734 |
. . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . 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 4342 |
. . . . . . . . . . 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 4342 |
. . . . . . . . . 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 4353 |
. . . . . . . . 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 793 |
. . . . . . . . . . 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 507 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ (ℤ ∖ {0})
→ (𝑏 ∈ ℤ
∧ 𝑏 ≠
0)) |
43 | | nnabscl 14449 |
. . . . . . . . . . 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 807 |
. . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . 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 578 |
. . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . 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 11721 |
. . . . . . . . . . . . . 14
⊢ (𝑐 ∈ ℕ ↔ (𝑐 ∈ ℤ ∧ 0 <
𝑐)) |
53 | 50, 51, 52 | sylanbrc 578 |
. . . . . . . . . . . . 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 4542 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ (ℤ ∖ {0})
→ 𝑐 ≠
0) |
55 | 54 | ad5antlr 730 |
. . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . 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 3961 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ (ℤ ∖ {0})
→ 𝑐 ∈
ℤ) |
58 | 57 | ad5antlr 730 |
. . . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . . 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 4342 |
. . . . . . . . . . . 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 4342 |
. . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . 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 578 |
. . . . . . . . . . . . 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 730 |
. . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . 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 730 |
. . . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . . 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 4342 |
. . . . . . . . . . . 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 730 |
. . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . 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 730 |
. . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . 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 4342 |
. . . . . . . . . . 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 4342 |
. . . . . . . . . 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 4353 |
. . . . . . . . 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 793 |
. . . . . . . . . . . 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 3810 |
. . . . . . . . . . 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 14449 |
. . . . . . . . . . 11
⊢ ((𝑐 ∈ ℤ ∧ 𝑐 ≠ 0) → (abs‘𝑐) ∈
ℕ) |
82 | 79, 80, 81 | syl2anc 579 |
. . . . . . . . . 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 807 |
. . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . . 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 12019 |
. . . . . . . . . . . . . . . . . . 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 11685 |
. . . . . . . . . . . . . . . . 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 13326 |
. . . . . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . . 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 13326 |
. . . . . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . . . . . 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 793 |
. . . . . . . . . . . . . . . . . 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 38063 |
. . . . . . . . . . . . . . . . 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 224 |
. . . . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . . . . 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 38063 |
. . . . . . . . . . . . . . . . 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 224 |
. . . . . . . . . . . . . . . 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 10934 |
. . . . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . 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 4901 |
. . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . 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 38063 |
. . . . . . . . . . . . . 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 249 |
. . . . . . . . . . . . 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 578 |
. . . . . . . . . . . 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 819 |
. . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 793 |
. . . . . . . . . . . . . 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 578 |
. . . . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . . 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 4342 |
. . . . . . . . . . . 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 4342 |
. . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . . 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 578 |
. . . . . . . . . . . . 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 819 |
. . . . . . . . . . . . . . 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 793 |
. . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . . 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 4342 |
. . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . . . 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 11685 |
. . . . . . . . . . . . . . . . . 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 13326 |
. . . . . . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . . . 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 13326 |
. . . . . . . . . . . . . . . . 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 10393 |
. . . . . . . . . . . . . . . 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 10367 |
. . . . . . . . . . . . . . . 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 3004 |
. . . . . . . . . . . . . . . . . . . . . 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 11716 |
. . . . . . . . . . . . . . . . . . . . . . 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 13191 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ((𝑎↑𝑛) = 0 ↔ 𝑎 = 0)) |
150 | 148, 136,
149 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . . . . . . . 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 785 |
. . . . . . . . . . . . . . . . . . . . 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 793 |
. . . . . . . . . . . . . . . . . . . . . 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 38063 |
. . . . . . . . . . . . . . . . . . . . 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 316 |
. . . . . . . . . . . . . . . . . . . 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 1011 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
((𝑎↑𝑛) = 0 ∨ 0 < (𝑎↑𝑛)) ↔ (¬ (𝑎↑𝑛) = 0 ∧ ¬ 0 < (𝑎↑𝑛))) |
157 | 151, 155,
156 | sylanbrc 578 |
. . . . . . . . . . . . . . . . . . 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 10501 |
. . . . . . . . . . . . . . . . . . 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 249 |
. . . . . . . . . . . . . . . . . 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 11716 |
. . . . . . . . . . . . . . . . . . . . . . 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 11985 |
. . . . . . . . . . . . . . . . . . . . . . 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 13315 |
. . . . . . . . . . . . . . . . . . . . 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 3004 |
. . . . . . . . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . . . . . . . . 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 38063 |
. . . . . . . . . . . . . . . . . . . . 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 316 |
. . . . . . . . . . . . . . . . . . . 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 1011 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
((𝑏↑𝑛) = 0 ∨ 0 < (𝑏↑𝑛)) ↔ (¬ (𝑏↑𝑛) = 0 ∧ ¬ 0 < (𝑏↑𝑛))) |
171 | 166, 169,
170 | sylanbrc 578 |
. . . . . . . . . . . . . . . . . . 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 10501 |
. . . . . . . . . . . . . . . . . . 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 249 |
. . . . . . . . . . . . . . . . . 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 10982 |
. . . . . . . . . . . . . . . . 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 10537 |
. . . . . . . . . . . . . . . . 17
⊢ (0 + 0) =
0 |
176 | 174, 175 | syl6breq 4916 |
. . . . . . . . . . . . . . . 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 10512 |
. . . . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . . . 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 2831 |
. . . . . . . . . . . . . . . 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 4887 |
. . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . 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 38063 |
. . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . 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 38052 |
. . . . . . . . . . . 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 4342 |
. . . . . . . . . . 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 4342 |
. . . . . . . . . 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 4342 |
. . . . . . . . 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 6917 |
. . . . . . . . . . . 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 6925 |
. . . . . . . . . . 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 2827 |
. . . . . . . . . 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 475 |
. . . . . . . . 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 6917 |
. . . . . . . . . . . 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 6926 |
. . . . . . . . . . 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 2827 |
. . . . . . . . . 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 475 |
. . . . . . . . 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 6917 |
. . . . . . . . . . 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 2835 |
. . . . . . . . . 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 475 |
. . . . . . . . 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 785 |
. . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . . 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 14426 |
. . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . 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 14559 |
. . . . . . . . . . . . . . . . 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 10392 |
. . . . . . . . . . . . . . . 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 479 |
. . . . . . . . . . . . . . . . 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 11685 |
. . . . . . . . . . . . . . . 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 11644 |
. . . . . . . . . . . . . . . . 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 13312 |
. . . . . . . . . . . . . . 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 13312 |
. . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . 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 805 |
. . . . . . . . . . . . . . . . . 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 11366 |
. . . . . . . . . . . . . . . . . 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 11436 |
. . . . . . . . . . . . . . . . 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 11469 |
. . . . . . . . . . . . . . . . . 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 11136 |
. . . . . . . . . . . . . . . 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 2831 |
. . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . . . . . 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 3810 |
. . . . . . . . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . . 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 14426 |
. . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . 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 14559 |
. . . . . . . . . . . . . . . . 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 10392 |
. . . . . . . . . . . . . . . 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 13312 |
. . . . . . . . . . . . . . 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 13312 |
. . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . 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 6928 |
. . . . . . . . . . . 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 11817 |
. . . . . . . . . . . . . . . 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 14426 |
. . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . 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 11716 |
. . . . . . . . . . . . . . . . . 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 14559 |
. . . . . . . . . . . . . . . 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 10392 |
. . . . . . . . . . . . . . 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 13312 |
. . . . . . . . . . . . . 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 13312 |
. . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . 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 2871 |
. . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . 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 6928 |
. . . . . . . . . . . 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 475 |
. . . . . . . . . . 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 4314 |
. . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . 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 475 |
. . . . . . . . . . 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 2871 |
. . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)) = 𝑎) |
271 | 270 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) = (𝑎↑𝑛)) |
272 | | iftrue 4314 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)) = 𝑏) |
273 | 272 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛) = (𝑏↑𝑛)) |
274 | 271, 273 | oveq12d 6928 |
. . . . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)) = 𝑐) |
277 | 276 | oveq1d 6925 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑏 → (if(0 < 𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛) = (𝑐↑𝑛)) |
278 | 277 | adantl 475 |
. . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . . . . . . . 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 817 |
. . . . . . . . . . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . . . . . . . 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 11431 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℕ |
286 | | nndivdvds 15373 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ ℕ ∧ 2 ∈
ℕ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℕ)) |
287 | 283, 285,
286 | sylancl 580 |
. . . . . . . . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . . . . . . . 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 15450 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑏 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2
∥ 𝑛) → (-𝑏↑𝑛) = -(𝑏↑𝑛)) |
290 | 281, 283,
288, 289 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . . . . 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 11633 |
. . . . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . . 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 10707 |
. . . . . . . . . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . 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 10564 |
. . . . . . . . . . . . . . . . . . 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 10726 |
. . . . . . . . . . . . . . . . . . 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 2861 |
. . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . . . . . . 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 2831 |
. . . . . . . . . . . . . . . . . . 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 10773 |
. . . . . . . . . . . . . . . . . 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 2865 |
. . . . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 <
𝑐 → if(0 < 𝑐, -𝑏, 𝑎) = -𝑏) |
309 | 308 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → (if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) = (-𝑏↑𝑛)) |
310 | | iftrue 4314 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 <
𝑐 → if(0 < 𝑐, 𝑐, -𝑐) = 𝑐) |
311 | 310 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛) = (𝑐↑𝑛)) |
312 | 309, 311 | oveq12d 6928 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → ((if(0 < 𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((-𝑏↑𝑛) + (𝑐↑𝑛))) |
313 | 312 | adantl 475 |
. . . . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → if(0 < 𝑐, 𝑎, -𝑏) = 𝑎) |
315 | 314 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛) = (𝑎↑𝑛)) |
316 | 315 | adantl 475 |
. . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . 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 819 |
. . . . . . . . . . . . . . . . . . . . . 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 740 |
. . . . . . . . . . . . . . . . . . . . . 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 11685 |
. . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . 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 10726 |
. . . . . . . . . . . . . . . . . . 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 10720 |
. . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . . 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 10707 |
. . . . . . . . . . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . . . . . . . . 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 10772 |
. . . . . . . . . . . . . . . . . . . . 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 10723 |
. . . . . . . . . . . . . . . . . . . . 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 10726 |
. . . . . . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . . . . . 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 10567 |
. . . . . . . . . . . . . . . . . . 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 2861 |
. . . . . . . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . . . . . . . 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 580 |
. . . . . . . . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . . . . . . . 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 15450 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2
∥ 𝑛) → (-𝑐↑𝑛) = -(𝑐↑𝑛)) |
342 | 324, 320,
340, 341 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . . . . . 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 1494 |
. . . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, -𝑏, 𝑎) = 𝑎) |
347 | 346 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, -𝑏, 𝑎)↑𝑛) = (𝑎↑𝑛)) |
348 | | iffalse 4317 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, 𝑐, -𝑐) = -𝑐) |
349 | 348 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, 𝑐, -𝑐)↑𝑛) = (-𝑐↑𝑛)) |
350 | 347, 349 | oveq12d 6928 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → ((if(0 <
𝑐, -𝑏, 𝑎)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((𝑎↑𝑛) + (-𝑐↑𝑛))) |
351 | 350 | adantl 475 |
. . . . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, 𝑎, -𝑏) = -𝑏) |
353 | 352 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, 𝑎, -𝑏)↑𝑛) = (-𝑏↑𝑛)) |
354 | 353 | adantl 475 |
. . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . 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 847 |
. . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎)) = if(0 < 𝑐, -𝑏, 𝑎)) |
358 | 357 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, 𝑎, if(0 < 𝑐, -𝑏, 𝑎))↑𝑛) = (if(0 < 𝑐, -𝑏, 𝑎)↑𝑛)) |
359 | | iffalse 4317 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐)) = if(0 < 𝑐, 𝑐, -𝑐)) |
360 | 359 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, 𝑏, if(0 < 𝑐, 𝑐, -𝑐))↑𝑛) = (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) |
361 | 358, 360 | oveq12d 6928 |
. . . . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏)) = if(0 < 𝑐, 𝑎, -𝑏)) |
364 | 363 | oveq1d 6925 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, 𝑐, if(0 < 𝑐, 𝑎, -𝑏))↑𝑛) = (if(0 < 𝑐, 𝑎, -𝑏)↑𝑛)) |
365 | 364 | adantl 475 |
. . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . 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 847 |
. . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . 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 6928 |
. . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . 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 819 |
. . . . . . . . . . . . . . . . . . . . 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 740 |
. . . . . . . . . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . . . . . . . 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 580 |
. . . . . . . . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . . . . . . . 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 15450 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2
∥ 𝑛) → (-𝑎↑𝑛) = -(𝑎↑𝑛)) |
385 | 379, 380,
383, 384 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . . . . 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 11685 |
. . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . 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 10707 |
. . . . . . . . . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . 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 10383 |
. . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . 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 10710 |
. . . . . . . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . . . . . . . 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 10386 |
. . . . . . . . . . . . . . . . . . . . 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 10563 |
. . . . . . . . . . . . . . . . . . . . 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 2869 |
. . . . . . . . . . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . . . . . . 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 2864 |
. . . . . . . . . . . . . . . . . . 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 10567 |
. . . . . . . . . . . . . . . . . 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 2861 |
. . . . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 <
𝑐 → if(0 < 𝑐, -𝑎, 𝑏) = -𝑎) |
406 | 405 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → (if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) = (-𝑎↑𝑛)) |
407 | 406, 311 | oveq12d 6928 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → ((if(0 < 𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((-𝑎↑𝑛) + (𝑐↑𝑛))) |
408 | 407 | adantl 475 |
. . . . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . . . 19
⊢ (0 <
𝑐 → if(0 < 𝑐, 𝑏, -𝑎) = 𝑏) |
410 | 409 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑐 → (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛) = (𝑏↑𝑛)) |
411 | 410 | adantl 475 |
. . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . 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 815 |
. . . . . . . . . . . . . . . . . . . . . . . 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 817 |
. . . . . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . . . 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 10707 |
. . . . . . . . . . . . . . . . . . . . 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 811 |
. . . . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . . 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 10564 |
. . . . . . . . . . . . . . . . . . . 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 10726 |
. . . . . . . . . . . . . . . . . . . 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 807 |
. . . . . . . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . . . . . . . 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 13309 |
. . . . . . . . . . . . . . . . . . . . . 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 10721 |
. . . . . . . . . . . . . . . . . . . . 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 2863 |
. . . . . . . . . . . . . . . . . . . 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 2865 |
. . . . . . . . . . . . . . . . . . 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 10602 |
. . . . . . . . . . . . . . . . . 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 10711 |
. . . . . . . . . . . . . . . . . . . . 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 2831 |
. . . . . . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . . . . . 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 803 |
. . . . . . . . . . . . . . . . . . . . . 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 580 |
. . . . . . . . . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . . . . . . . . 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 1494 |
. . . . . . . . . . . . . . . . . . . 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 6926 |
. . . . . . . . . . . . . . . . . . 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 10733 |
. . . . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . . . 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 1494 |
. . . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, -𝑎, 𝑏) = 𝑏) |
446 | 445 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, -𝑎, 𝑏)↑𝑛) = (𝑏↑𝑛)) |
447 | 446, 349 | oveq12d 6928 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → ((if(0 <
𝑐, -𝑎, 𝑏)↑𝑛) + (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) = ((𝑏↑𝑛) + (-𝑐↑𝑛))) |
448 | 447 | adantl 475 |
. . . . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬ 0
< 𝑐 → if(0 <
𝑐, 𝑏, -𝑎) = -𝑎) |
450 | 449 | oveq1d 6925 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑐 → (if(0 <
𝑐, 𝑏, -𝑎)↑𝑛) = (-𝑎↑𝑛)) |
451 | 450 | adantl 475 |
. . . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . . 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 847 |
. . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎) = if(0 < 𝑐, -𝑎, 𝑏)) |
455 | 454 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) = (if(0 < 𝑐, -𝑎, 𝑏)↑𝑛)) |
456 | | iftrue 4314 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 <
𝑏 → if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏) = if(0 < 𝑐, 𝑐, -𝑐)) |
457 | 456 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → (if(0 < 𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛) = (if(0 < 𝑐, 𝑐, -𝑐)↑𝑛)) |
458 | 455, 457 | oveq12d 6928 |
. . . . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . . . 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 4314 |
. . . . . . . . . . . . . . . . 17
⊢ (0 <
𝑏 → if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐) = if(0 < 𝑐, 𝑏, -𝑎)) |
461 | 460 | oveq1d 6925 |
. . . . . . . . . . . . . . . 16
⊢ (0 <
𝑏 → (if(0 < 𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛) = (if(0 < 𝑐, 𝑏, -𝑎)↑𝑛)) |
462 | 461 | adantl 475 |
. . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . 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 10602 |
. . . . . . . . . . . . . . . 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 580 |
. . . . . . . . . . . . . . . . . . . 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 317 |
. . . . . . . . . . . . . . . . . . 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 1494 |
. . . . . . . . . . . . . . . . . 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 1494 |
. . . . . . . . . . . . . . . . . 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 6928 |
. . . . . . . . . . . . . . . . 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 13314 |
. . . . . . . . . . . . . . . . . 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 13314 |
. . . . . . . . . . . . . . . . . 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 10733 |
. . . . . . . . . . . . . . . . 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 2864 |
. . . . . . . . . . . . . . . 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 1494 |
. . . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎) = -𝑎) |
479 | 478 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, if(0 < 𝑐, -𝑎, 𝑏), -𝑎)↑𝑛) = (-𝑎↑𝑛)) |
480 | | iffalse 4317 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏) = -𝑏) |
481 | 480 | oveq1d 6925 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, if(0 < 𝑐, 𝑐, -𝑐), -𝑏)↑𝑛) = (-𝑏↑𝑛)) |
482 | 479, 481 | oveq12d 6928 |
. . . . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . . 17
⊢ (¬ 0
< 𝑏 → if(0 <
𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐) = -𝑐) |
485 | 484 | oveq1d 6925 |
. . . . . . . . . . . . . . . 16
⊢ (¬ 0
< 𝑏 → (if(0 <
𝑏, if(0 < 𝑐, 𝑏, -𝑎), -𝑐)↑𝑛) = (-𝑐↑𝑛)) |
486 | 485 | adantl 475 |
. . . . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . . . 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 847 |
. . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . . 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 6928 |
. . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . . 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 475 |
. . . . . . . . . . . . 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 2871 |
. . . . . . . . . . . 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 847 |
. . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . 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 4317 |
. . . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . . 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 6928 |
. . . . . . . . . . . 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 475 |
. . . . . . . . . . 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 4317 |
. . . . . . . . . . . . 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 6925 |
. . . . . . . . . . . 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 475 |
. . . . . . . . . . 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 2871 |
. . . . . . . . . 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 847 |
. . . . . . . . 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 38034 |
. . . . . . . 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 3243 |
. . . . . . 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 3240 |
. . . . . 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 3240 |
. . . . 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 3217 |
. . . 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 3003 |
. . . . . . . . . . . . 13
⊢ (¬
((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛)) |
517 | 516 | bicomi 216 |
. . . . . . . . . . . 12
⊢ (((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
518 | 517 | rexbii 3251 |
. . . . . . . . . . 11
⊢
(∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ∃𝑐 ∈ (ℤ ∖ {0}) ¬ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
519 | | rexnal 3203 |
. . . . . . . . . . 11
⊢
(∃𝑐 ∈
(ℤ ∖ {0}) ¬ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ¬ ∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
520 | 518, 519 | bitri 267 |
. . . . . . . . . 10
⊢
(∃𝑐 ∈
(ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) = (𝑐↑𝑛) ↔ ¬ ∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) |
521 | 520 | rexbii 3251 |
. . . . . . . . 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 3203 |
. . . . . . . . 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 267 |
. . . . . . . 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 3251 |
. . . . . . 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 3203 |
. . . . . . 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 267 |
. . . . . 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 3251 |
. . . . 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 3203 |
. . . . 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 267 |
. . . 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 3003 |
. . . . . . . . . . . . 13
⊢ (¬
((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛)) |
531 | 530 | bicomi 216 |
. . . . . . . . . . . 12
⊢ (((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
532 | 531 | rexbii 3251 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ∃𝑧 ∈ ℕ ¬ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
533 | | rexnal 3203 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
ℕ ¬ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ¬ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
534 | 532, 533 | bitri 267 |
. . . . . . . . . 10
⊢
(∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
535 | 534 | rexbii 3251 |
. . . . . . . . 9
⊢
(∃𝑦 ∈
ℕ ∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ∃𝑦 ∈ ℕ ¬ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
536 | | rexnal 3203 |
. . . . . . . . 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 267 |
. . . . . . . 8
⊢
(∃𝑦 ∈
ℕ ∃𝑧 ∈
ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) = (𝑧↑𝑛) ↔ ¬ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛)) |
538 | 537 | rexbii 3251 |
. . . . . . 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 3203 |
. . . . . . 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 267 |
. . . . . 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 3251 |
. . . . 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 3203 |
. . . . 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 267 |
. . . 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 283 |
. . 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 11394 |
. . . . . . . 8
⊢ ¬ 0
∈ ℕ |
547 | | disjsn 4467 |
. . . . . . . 8
⊢ ((ℕ
∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) |
548 | 546, 547 | mpbir 223 |
. . . . . . 7
⊢ (ℕ
∩ {0}) = ∅ |
549 | | disj3 4247 |
. . . . . . 7
⊢ ((ℕ
∩ {0}) = ∅ ↔ ℕ = (ℕ ∖ {0})) |
550 | 548, 549 | mpbi 222 |
. . . . . 6
⊢ ℕ =
(ℕ ∖ {0}) |
551 | | nnssz 11731 |
. . . . . . 7
⊢ ℕ
⊆ ℤ |
552 | | ssdif 3974 |
. . . . . . 7
⊢ (ℕ
⊆ ℤ → (ℕ ∖ {0}) ⊆ (ℤ ∖
{0})) |
553 | 551, 552 | ax-mp 5 |
. . . . . 6
⊢ (ℕ
∖ {0}) ⊆ (ℤ ∖ {0}) |
554 | 550, 553 | eqsstri 3860 |
. . . . 5
⊢ ℕ
⊆ (ℤ ∖ {0}) |
555 | | ssel 3821 |
. . . . . . 7
⊢ (ℕ
⊆ (ℤ ∖ {0}) → (𝑎 ∈ ℕ → 𝑎 ∈ (ℤ ∖
{0}))) |
556 | | ssel 3821 |
. . . . . . . . 9
⊢ (ℕ
⊆ (ℤ ∖ {0}) → (𝑏 ∈ ℕ → 𝑏 ∈ (ℤ ∖
{0}))) |
557 | | ssralv 3891 |
. . . . . . . . 9
⊢ (ℕ
⊆ (ℤ ∖ {0}) → (∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) → ∀𝑐 ∈ ℕ ((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))) |
558 | 556, 557 | imim12d 81 |
. . . . . . . 8
⊢ (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)))) |
559 | 558 | ralimdv2 3170 |
. . . . . . 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))) |
560 | 555, 559 | 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)))) |
561 | 560 | ralimdv2 3170 |
. . . . 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))) |
562 | 554, 561 | 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)) |
563 | | oveq1 6917 |
. . . . . . 7
⊢ (𝑎 = 𝑥 → (𝑎↑𝑛) = (𝑥↑𝑛)) |
564 | 563 | oveq1d 6925 |
. . . . . 6
⊢ (𝑎 = 𝑥 → ((𝑎↑𝑛) + (𝑏↑𝑛)) = ((𝑥↑𝑛) + (𝑏↑𝑛))) |
565 | 564 | neeq1d 3058 |
. . . . 5
⊢ (𝑎 = 𝑥 → (((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑥↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))) |
566 | | oveq1 6917 |
. . . . . . 7
⊢ (𝑏 = 𝑦 → (𝑏↑𝑛) = (𝑦↑𝑛)) |
567 | 566 | oveq2d 6926 |
. . . . . 6
⊢ (𝑏 = 𝑦 → ((𝑥↑𝑛) + (𝑏↑𝑛)) = ((𝑥↑𝑛) + (𝑦↑𝑛))) |
568 | 567 | neeq1d 3058 |
. . . . 5
⊢ (𝑏 = 𝑦 → (((𝑥↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑐↑𝑛))) |
569 | | oveq1 6917 |
. . . . . 6
⊢ (𝑐 = 𝑧 → (𝑐↑𝑛) = (𝑧↑𝑛)) |
570 | 569 | neeq2d 3059 |
. . . . 5
⊢ (𝑐 = 𝑧 → (((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑐↑𝑛) ↔ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛))) |
571 | 565, 568,
570 | cbvral3v 3393 |
. . . 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)) |
572 | 562, 571 | sylib 210 |
. . 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)) |
573 | 572 | ralimi 3161 |
. 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)) |
574 | 545, 573 | impbii 201 |
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)) |