Metamath Proof Explorer

Theorem dffltz

Description: Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes ( a ^ n ) + ( b ^ n ) = ( c ^ n ) , and adds the negative of any negative term to both sides, thus creating the corresponding equation with only positive integers. There are six combinations of negativity, so the proof is particularly long. (Contributed by Steven Nguyen, 27-Feb-2023)

Ref Expression
Assertion dffltz ( ∀ 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 ) )

Proof

Step Hyp Ref Expression
1 simp-4r ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
2 eldifi ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → 𝑎 ∈ ℤ )
3 eldifsni ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → 𝑎 ≠ 0 )
4 2 3 jca ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ( 𝑎 ∈ ℤ ∧ 𝑎 ≠ 0 ) )
5 nnabscl ⊢ ( ( 𝑎 ∈ ℤ ∧ 𝑎 ≠ 0 ) → ( abs ‘ 𝑎 ) ∈ ℕ )
6 1 4 5 3syl ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( abs ‘ 𝑎 ) ∈ ℕ )
7 simp-6r ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( 𝑎 ∈ ℕ ↔ ( 𝑎 ∈ ℤ ∧ 0 < 𝑎 ) )
11 8 9 10 sylanbrc ⊢ ( ( ( ( ( ( ( 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 ⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → 𝑏 ≠ 0 )
13 12 ad6antlr ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → 𝑏 ∈ ℤ )
16 15 ad6antlr ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( 𝑏 ∈ ℕ ↔ ( 𝑏 ∈ ℤ ∧ 0 < 𝑏 ) )
32 29 30 31 sylanbrc ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → ( 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0 ) )
43 nnabscl ⊢ ( ( 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0 ) → ( abs ‘ 𝑏 ) ∈ ℕ )
44 41 42 43 3syl ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( abs ‘ 𝑏 ) ∈ ℕ )
45 simp-5r ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( 𝑐 ∈ ℕ ↔ ( 𝑐 ∈ ℤ ∧ 0 < 𝑐 ) )
53 50 51 52 sylanbrc ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( 𝑐 ∈ ( ℤ ∖ { 0 } ) → 𝑐 ≠ 0 )
55 54 ad5antlr ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( 𝑐 ∈ ( ℤ ∖ { 0 } ) → 𝑐 ∈ ℤ )
58 57 ad5antlr ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( 𝑐 ∈ ℤ ∧ 𝑐 ≠ 0 ) → ( abs ‘ 𝑐 ) ∈ ℕ )
82 79 80 81 syl2anc ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → 𝑛 ∈ ℕ )
89 88 ad7antr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ¬ 𝑎 = 0 )
146 133 145 syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 𝑎 ∈ ℤ → 𝑎 ∈ ℂ )
148 133 2 147 3syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( 𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ) → ( ( 𝑎 ↑ 𝑛 ) = 0 ↔ 𝑎 = 0 ) )
150 148 136 149 syl2anc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ¬ ( ( 𝑎 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑎 ↑ 𝑛 ) ) ↔ ( ¬ ( 𝑎 ↑ 𝑛 ) = 0 ∧ ¬ 0 < ( 𝑎 ↑ 𝑛 ) ) )
157 151 155 156 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 𝑏 ∈ ℤ → 𝑏 ∈ ℂ )
161 139 15 160 3syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → 𝑛 ∈ ℤ )
164 163 ad7antr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ¬ ( ( 𝑏 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑏 ↑ 𝑛 ) ) ↔ ( ¬ ( 𝑏 ↑ 𝑛 ) = 0 ∧ ¬ 0 < ( 𝑏 ↑ 𝑛 ) ) )
171 166 169 170 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 0 + 0 ) = 0
176 174 175 breqtrdi ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 𝑎 ∈ ℝ → ( ( abs ‘ 𝑎 ) ↑ 2 ) = ( 𝑎 ↑ 2 ) )
206 204 205 syl ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ 2 ∈ ℕ0
214 213 a1i ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ )
220 218 88 219 3syl ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ 2 ≠ 0
223 222 a1i ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 𝑏 ∈ ℝ → ( ( abs ‘ 𝑏 ) ↑ 2 ) = ( 𝑏 ↑ 2 ) )
233 231 232 syl ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 𝑐 ∈ ℝ → ( ( abs ‘ 𝑐 ) ↑ 2 ) = ( 𝑐 ↑ 2 ) )
247 245 246 syl ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 𝑐 ∈ ℤ → 𝑐 ∈ ℂ )
250 78 57 249 3syl ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) = 𝑎 )
271 270 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
272 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) = 𝑏 )
273 272 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
274 271 273 oveq12d ⊢ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) = 𝑐 )
277 276 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
278 277 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ 2 ∈ ℕ
286 nndivdvds ⊢ ( ( 𝑛 ∈ ℕ ∧ 2 ∈ ℕ ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
287 283 285 286 sylancl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( 𝑏 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑏 ↑ 𝑛 ) = - ( 𝑏 ↑ 𝑛 ) )
290 281 283 288 289 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 )
293 282 88 292 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) = - 𝑏 )
309 308 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
310 iftrue ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) = 𝑐 )
311 310 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
312 309 311 oveq12d ⊢ ( 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
313 312 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) = 𝑎 )
315 314 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
316 315 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( 𝑐 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑐 ↑ 𝑛 ) = - ( 𝑐 ↑ 𝑛 ) )
342 324 320 340 341 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) = 𝑎 )
347 346 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
348 iffalse ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) = - 𝑐 )
349 348 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
350 347 349 oveq12d ⊢ ( ¬ 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
351 350 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) = - 𝑏 )
353 352 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
354 353 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) = if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) )
358 357 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) )
359 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) = if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) )
360 359 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) )
361 358 360 oveq12d ⊢ ( ¬ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) = if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) )
364 363 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
365 364 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( 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 ⊢ ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( 𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑎 ↑ 𝑛 ) = - ( 𝑎 ↑ 𝑛 ) )
385 379 380 383 384 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) = - 𝑎 )
406 405 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
407 406 311 oveq12d ⊢ ( 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
408 407 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) = 𝑏 )
410 409 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
411 410 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) = 𝑏 )
446 445 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
447 446 349 oveq12d ⊢ ( ¬ 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
448 447 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) = - 𝑎 )
450 449 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
451 450 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) = if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) )
455 454 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) )
456 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) = if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) )
457 456 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) )
458 455 457 oveq12d ⊢ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) = if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) )
461 460 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
462 461 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) = - 𝑎 )
479 478 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
480 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) = - 𝑏 )
481 480 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
482 479 481 oveq12d ⊢ ( ¬ 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) = - 𝑐 )
485 484 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
486 485 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ¬ 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 ⊢ ( ¬ 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 ⊢ ( ¬ 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 ⊢ ( ¬ 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 ⊢ ( ¬ 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ¬ 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 ⊢ ( ¬ 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ¬ ( 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 ⊢ ( ¬ ( 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 ⊢ ( ¬ ( 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 ⊢ ( ¬ ( 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 ⊢ ( ¬ ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ¬ ( 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 ⊢ ( ¬ ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( ( ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( 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 ⊢ ( 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 ⊢ ( ∃ 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 ⊢ ( ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
517 516 bicomi ⊢ ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
518 517 rexbii ⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
519 rexnal ⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
520 518 519 bitri ⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
521 520 rexbii ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) )
531 530 bicomi ⊢ ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
532 531 rexbii ⊢ ( ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑧 ∈ ℕ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
533 rexnal ⊢ ( ∃ 𝑧 ∈ ℕ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
534 532 533 bitri ⊢ ( ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
535 534 rexbii ⊢ ( ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑦 ∈ ℕ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
536 rexnal ⊢ ( ∃ 𝑦 ∈ ℕ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
537 535 536 bitri ⊢ ( ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
538 537 rexbii ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ∃ 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 ⊢ ( ¬ ∀ 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 ⊢ ( ∀ 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 dfn2 ⊢ ℕ = ( ℕ0 ∖ { 0 } )
547 nn0ssz ⊢ ℕ0 ⊆ ℤ
548 ssdif ⊢ ( ℕ0 ⊆ ℤ → ( ℕ0 ∖ { 0 } ) ⊆ ( ℤ ∖ { 0 } ) )
549 547 548 ax-mp ⊢ ( ℕ0 ∖ { 0 } ) ⊆ ( ℤ ∖ { 0 } )
550 546 549 eqsstri ⊢ ℕ ⊆ ( ℤ ∖ { 0 } )
551 ssel ⊢ ( ℕ ⊆ ( ℤ ∖ { 0 } ) → ( 𝑎 ∈ ℕ → 𝑎 ∈ ( ℤ ∖ { 0 } ) ) )
552 ss2ralv ⊢ ( 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 ) ) )
553 551 552 imim12d ⊢ ( 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 ) ) ) )
554 553 ralimdv2 ⊢ ( 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 ) ) )
555 550 554 ax-mp ⊢ ( ∀ 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 ) )
556 oveq1 ⊢ ( 𝑎 = 𝑥 → ( 𝑎 ↑ 𝑛 ) = ( 𝑥 ↑ 𝑛 ) )
557 556 oveq1d ⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
558 557 neeq1d ⊢ ( 𝑎 = 𝑥 → ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
559 oveq1 ⊢ ( 𝑏 = 𝑦 → ( 𝑏 ↑ 𝑛 ) = ( 𝑦 ↑ 𝑛 ) )
560 559 oveq2d ⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) )
561 560 neeq1d ⊢ ( 𝑏 = 𝑦 → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
562 oveq1 ⊢ ( 𝑐 = 𝑧 → ( 𝑐 ↑ 𝑛 ) = ( 𝑧 ↑ 𝑛 ) )
563 562 neeq2d ⊢ ( 𝑐 = 𝑧 → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ) )
564 558 561 563 cbvral3vw ⊢ ( ∀ 𝑎 ∈ ℕ ∀ 𝑏 ∈ ℕ ∀ 𝑐 ∈ ℕ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
565 555 564 sylib ⊢ ( ∀ 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 ) )
566 565 ralimi ⊢ ( ∀ 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 ) )
567 545 566 impbii ⊢ ( ∀ 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 ) )