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		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 ) )
2	1	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 ) ) )
3	2	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 ) ) )
4		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 ) )
5	4	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 ) ) )
6	5	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 ) ) )
7		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 ) )
8	7	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 ) ) )
9		simp-4r	⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
10		eldifi	⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → 𝑎 ∈ ℤ )
11		eldifsni	⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → 𝑎 ≠ 0 )
12	10 11	jca	⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ( 𝑎 ∈ ℤ ∧ 𝑎 ≠ 0 ) )
13		nnabscl	⊢ ( ( 𝑎 ∈ ℤ ∧ 𝑎 ≠ 0 ) → ( abs ‘ 𝑎 ) ∈ ℕ )
14	9 12 13	3syl	⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( abs ‘ 𝑎 ) ∈ ℕ )
15		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 } ) )
16	15	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 )
17		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 )
18		elnnz	⊢ ( 𝑎 ∈ ℕ ↔ ( 𝑎 ∈ ℤ ∧ 0 < 𝑎 ) )
19	16 17 18	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 )
20		eldifsni	⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → 𝑏 ≠ 0 )
21	20	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 )
22		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 )
23		eldifi	⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → 𝑏 ∈ ℤ )
24	23	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 )
25	21 22 24	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 )
26		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 } ) )
27	26	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 )
28		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 )
29	27 28 18	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 )
30	25 29	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 )
31	19 30	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 )
32	11	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 )
33		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 )
34	10	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 )
35	32 33 34	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 )
36		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 } ) )
37	36	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 )
38		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 )
39		elnnz	⊢ ( 𝑏 ∈ ℕ ↔ ( 𝑏 ∈ ℤ ∧ 0 < 𝑏 ) )
40	37 38 39	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 )
41	35 40	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 )
42	11	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 )
43		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 )
44	10	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 )
45	42 43 44	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 )
46	41 45	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 )
47	31 46	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 )
48	14 47	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 )
49		simpllr	⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
50	23 20	jca	⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → ( 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0 ) )
51		nnabscl	⊢ ( ( 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0 ) → ( abs ‘ 𝑏 ) ∈ ℕ )
52	49 50 51	3syl	⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( abs ‘ 𝑏 ) ∈ ℕ )
53		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 } ) )
54	53	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 )
55		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 )
56	54 55 39	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 )
57		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 } ) )
58	57	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 )
59		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 )
60		elnnz	⊢ ( 𝑐 ∈ ℕ ↔ ( 𝑐 ∈ ℤ ∧ 0 < 𝑐 ) )
61	58 59 60	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 )
62		eldifsni	⊢ ( 𝑐 ∈ ( ℤ ∖ { 0 } ) → 𝑐 ≠ 0 )
63	62	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 )
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		eldifi	⊢ ( 𝑐 ∈ ( ℤ ∖ { 0 } ) → 𝑐 ∈ ℤ )
66	65	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 )
67	63 64 66	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 )
68	61 67	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 )
69	56 68	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 )
70		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 } ) )
71	70	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 )
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 < 𝑐 ) → 0 < 𝑐 )
73	71 72 60	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 )
74	62	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 )
75		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 )
76	65	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 )
77	74 75 76	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 )
78	73 77	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 )
79	20	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 )
80		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 )
81	23	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 )
82	79 80 81	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 )
83	78 82	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 )
84	69 83	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 )
85	52 84	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 )
86		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 } ) )
87	86	eldifad	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ℤ )
88	86 62	syl	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ≠ 0 )
89		nnabscl	⊢ ( ( 𝑐 ∈ ℤ ∧ 𝑐 ≠ 0 ) → ( abs ‘ 𝑐 ) ∈ ℕ )
90	87 88 89	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 )
91		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 } ) )
92	91	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 )
93		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 } ) )
94	93	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 )
95	94	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 )
96		eluz3nn	⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → 𝑛 ∈ ℕ )
97	96	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 )
98	97	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 )
99	95 98	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 )
100		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 } ) )
101	100	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 )
102	101	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 )
103	102 98	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 )
104		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 )
105		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 )
106	95 97 105	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 ) ) )
107	104 106	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 ) )
108		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 )
109	102 97 105	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 ) ) )
110	108 109	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 ) )
111	99 103 107 110	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 ) ) )
112		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 ) )
113	111 112	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 ) )
114	92	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 )
115	114 97 105	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 ) ) )
116	113 115	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 )
117	92 116 60	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 )
118		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 } ) )
119	118	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 )
120		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 )
121	119 120 18	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 )
122		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 } ) )
123	122 20	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 )
124		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 )
125	122	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 )
126	123 124 125	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 )
127	121 126	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 )
128	117 127	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 )
129		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 } ) )
130	129	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 )
131		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 )
132	130 131 39	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 )
133		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 } ) )
134	133 11	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 )
135		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 )
136	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 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ℤ )
137	134 135 136	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 )
138	132 137	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 )
139		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 } ) )
140	139 62	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 )
141		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 } ) )
142	141	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 )
143	142	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 )
144	96	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 )
145	144	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 )
146	143 145	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 )
147		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 } ) )
148	147	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 )
149	148	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 )
150	149 145	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 )
151	146 150	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 )
152		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 )
153	11	neneqd	⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ¬ 𝑎 = 0 )
154	141 153	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 )
155		zcn	⊢ ( 𝑎 ∈ ℤ → 𝑎 ∈ ℂ )
156	141 10 155	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 )
157		expeq0	⊢ ( ( 𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ) → ( ( 𝑎 ↑ 𝑛 ) = 0 ↔ 𝑎 = 0 ) )
158	156 144 157	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 ) )
159	154 158	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 )
160		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 )
161		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 )
162	143 144 161	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 ) ) )
163	160 162	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 ) )
164		ioran	⊢ ( ¬ ( ( 𝑎 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑎 ↑ 𝑛 ) ) ↔ ( ¬ ( 𝑎 ↑ 𝑛 ) = 0 ∧ ¬ 0 < ( 𝑎 ↑ 𝑛 ) ) )
165	159 163 164	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 ) ) )
166	146 152	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 ) ) ) )
167	165 166	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 )
168		zcn	⊢ ( 𝑏 ∈ ℤ → 𝑏 ∈ ℂ )
169	147 23 168	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 )
170	147 20	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 )
171		eluzelz	⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → 𝑛 ∈ ℤ )
172	171	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 )
173	169 170 172	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 )
174	173	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 )
175		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 )
176	149 144 161	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 ) ) )
177	175 176	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 ) )
178		ioran	⊢ ( ¬ ( ( 𝑏 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑏 ↑ 𝑛 ) ) ↔ ( ¬ ( 𝑏 ↑ 𝑛 ) = 0 ∧ ¬ 0 < ( 𝑏 ↑ 𝑛 ) ) )
179	174 177 178	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 ) ) )
180	150 152	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 ) ) ) )
181	179 180	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 )
182	146 150 152 152 167 181	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 ) )
183		00id	⊢ ( 0 + 0 ) = 0
184	182 183	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 )
185	151 152 184	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 ) ) )
186		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 ) )
187	186	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 ) ) )
188	187	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 ) ) ) )
189	185 188	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 ) )
190	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 ) → 𝑐 ∈ ℤ )
191	190	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 )
192	191 144 161	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 ) ) )
193	189 192	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 )
194	140 193 190	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 )
195	138 194	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 )
196	128 195	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 )
197	90 196	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 )
198		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 ) )
199		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 } ) )
200	199	eldifad	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ℤ )
201	200	zred	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ℝ )
202		absresq	⊢ ( 𝑎 ∈ ℝ → ( ( abs ‘ 𝑎 ) ↑ 2 ) = ( 𝑎 ↑ 2 ) )
203	201 202	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 ) )
204	203	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 ) ) )
205	199 10 155	3syl	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ℂ )
206	205	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 )
207	206	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 )
208		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 )
209	208	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 )
210		2nn0	⊢ 2 ∈ ℕ0
211	210	a1i	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 2 ∈ ℕ0 )
212	207 209 211	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 ) ) )
213	205 209 211	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 ) ) )
214	204 212 213	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 ) ) ) )
215		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 ) )
216		nncn	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ )
217	215 96 216	3syl	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑛 ∈ ℂ )
218		2cnd	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 2 ∈ ℂ )
219		2ne0	⊢ 2 ≠ 0
220	219	a1i	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 2 ≠ 0 )
221	217 218 220	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 )
222	221	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 ) ) )
223	222	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 ) ) ) )
224	222	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 ) ) ) )
225	214 223 224	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 ) )
226		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 } ) )
227	226	eldifad	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ℤ )
228	227	zred	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ℝ )
229		absresq	⊢ ( 𝑏 ∈ ℝ → ( ( abs ‘ 𝑏 ) ↑ 2 ) = ( 𝑏 ↑ 2 ) )
230	228 229	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 ) )
231	230	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 ) ) )
232	226 23 168	3syl	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ℂ )
233	232	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 )
234	233	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 )
235	234 209 211	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 ) ) )
236	232 209 211	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 ) ) )
237	231 235 236	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 ) ) ) )
238	222	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 ) ) ) )
239	222	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 ) ) ) )
240	237 238 239	3eqtr4d	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
241	225 240	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 ) ) )
242	87	zred	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ℝ )
243		absresq	⊢ ( 𝑐 ∈ ℝ → ( ( abs ‘ 𝑐 ) ↑ 2 ) = ( 𝑐 ↑ 2 ) )
244	242 243	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 ) )
245	244	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 ) ) )
246		zcn	⊢ ( 𝑐 ∈ ℤ → 𝑐 ∈ ℂ )
247	86 65 246	3syl	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ℂ )
248	247	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 )
249	248	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 )
250	249 209 211	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 ) ) )
251	247 209 211	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 ) ) )
252	245 250 251	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 ) ) ) )
253	222	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 ) ) ) )
254	222	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 ) ) ) )
255	252 253 254	3eqtr4d	⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑐 ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
256	198 241 255	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 ) )
257		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 ) )
258	257	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 ) )
259		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 ) )
260	259	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 ) )
261	258 260	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 ) ) )
262	261	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 ) ) )
263		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 ) )
264	263	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 ) )
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 ‘ 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 ) )
266	256 262 265	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 ) )
267		iftrue	⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) = 𝑎 )
268	267	oveq1d	⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
269		iftrue	⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) = 𝑏 )
270	269	oveq1d	⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
271	268 270	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 ) ) )
272	271	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 ) ) )
273		iftrue	⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) = 𝑐 )
274	273	oveq1d	⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( 𝑐 ↑ 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 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
276	112 272 275	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 ) )
277		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 } ) )
278	277 23 168	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 )
279		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 ) )
280	279 96	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 )
281		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 )
282		2nn	⊢ 2 ∈ ℕ
283		nndivdvds	⊢ ( ( 𝑛 ∈ ℕ ∧ 2 ∈ ℕ ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
284	280 282 283	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 ) )
285	281 284	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 )
286		oexpneg	⊢ ( ( 𝑏 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑏 ↑ 𝑛 ) = - ( 𝑏 ↑ 𝑛 ) )
287	278 280 285 286	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 ) )
288	287	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 ) ) )
289		nnnn0	⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 )
290	279 96 289	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 )
291	278 290	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 )
292	291	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 )
293		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 } ) )
294	293 65 246	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 )
295	294 290	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 )
296	292 295	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 ) ) )
297	295 291	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 ) ) )
298	296 297	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 ) ) )
299	118 10 155	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 )
300	299 290	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 )
301		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 ) )
302	301	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 ) ) )
303	300 291 302	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 ) )
304	288 298 303	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 ) )
305		iftrue	⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) = - 𝑏 )
306	305	oveq1d	⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
307		iftrue	⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) = 𝑐 )
308	307	oveq1d	⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
309	306 308	oveq12d	⊢ ( 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
310	309	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 ) ) )
311		iftrue	⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) = 𝑎 )
312	311	oveq1d	⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( 𝑎 ↑ 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 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
314	304 310 313	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 ) )
315		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 } ) )
316	315 10 155	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 )
317	96	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 )
318	317	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 )
319	316 318	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 )
320		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 } ) )
321	320 65 246	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 )
322	321 318	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 )
323	319 322	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 ) ) )
324	319 322	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 )
325	122 23 168	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 )
326	325 318	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 )
327	326	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 )
328		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 ) )
329	319 326 328	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 ) ) )
330	322 319	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 ) )
331	322 326	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 ) ) )
332	329 330 331	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 ) ) )
333	322 324 327 332	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 ) )
334	323 333	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 ) )
335		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 )
336	317 282 283	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 ) )
337	335 336	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 )
338		oexpneg	⊢ ( ( 𝑐 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑐 ↑ 𝑛 ) = - ( 𝑐 ↑ 𝑛 ) )
339	321 317 337 338	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 ) )
340	339	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 ) ) )
341	325 317 337 286	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 ) )
342	334 340 341	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 ) )
343		iffalse	⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) = 𝑎 )
344	343	oveq1d	⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
345		iffalse	⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) = - 𝑐 )
346	345	oveq1d	⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
347	344 346	oveq12d	⊢ ( ¬ 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
348	347	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 ) ) )
349		iffalse	⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) = - 𝑏 )
350	349	oveq1d	⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 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 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
352	342 348 351	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 ) )
353	314 352	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 ) )
354		iffalse	⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) = if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) )
355	354	oveq1d	⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) )
356		iffalse	⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) = if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) )
357	356	oveq1d	⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) )
358	355 357	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 ) ) )
359	358	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 ) ) )
360		iffalse	⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) = if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) )
361	360	oveq1d	⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 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 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
363	353 359 362	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 ) )
364	276 363	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 ) )
365		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 ) ) )
366	365	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 ) )
367		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 ) ) )
368	367	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 ) )
369	366 368	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 ) ) )
370	369	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 ) ) )
371		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 ) ) )
372	371	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 ) )
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 , 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 ) )
374	364 370 373	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 ) )
375		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 } ) )
376	375 10 155	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 )
377	96	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 )
378		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 )
379	377 282 283	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 ) )
380	378 379	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 )
381		oexpneg	⊢ ( ( 𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑎 ↑ 𝑛 ) = - ( 𝑎 ↑ 𝑛 ) )
382	376 377 380 381	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 ) )
383	382	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 ) ) )
384	377	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 )
385	376 384	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 )
386	385	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 )
387		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 } ) )
388	387 65 246	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 )
389	388 384	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 )
390	386 389	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 )
391	129 23 168	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 )
392	391 384	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 )
393	385	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 )
394	393	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 ) ) )
395	385 386 389	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 ) ) ) )
396	389	addlidd	⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 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 ) )
397	394 395 396	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 ) )
398		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 ) )
399	397 398	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 ) ) )
400	385 390 392 399	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 ) )
401	383 400	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 ) )
402		iftrue	⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) = - 𝑎 )
403	402	oveq1d	⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
404	403 308	oveq12d	⊢ ( 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
405	404	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 ) ) )
406		iftrue	⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) = 𝑏 )
407	406	oveq1d	⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( 𝑏 ↑ 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 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
409	401 405 408	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 ) )
410		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 } ) )
411	410 23 168	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 )
412		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 ) )
413	412 96 289	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 )
414	411 413	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 )
415	414	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 )
416		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 } ) )
417	416 65 246	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 )
418	417 413	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 )
419	415 418	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 ) ) )
420	418 414	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 ) ) )
421		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 ) )
422	421	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 ) ) )
423	133 10 155	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 )
424	423 413	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 )
425	424 414	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 ) )
426	422 425	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 ) )
427	419 420 426	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 ) )
428	427	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 ) )
429	414	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 ) )
430	429	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 ) )
431	430	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 ) ) )
432	412 96	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 )
433		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 )
434	432 282 283	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 ) )
435	433 434	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 )
436	417 432 435 338	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 ) )
437	436	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 ) ) )
438	415 418	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 ) ) )
439	431 437 438	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 ) ) )
440	423 432 435 381	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 ) )
441	428 439 440	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 ) )
442		iffalse	⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) = 𝑏 )
443	442	oveq1d	⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
444	443 346	oveq12d	⊢ ( ¬ 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
445	444	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 ) ) )
446		iffalse	⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) = - 𝑎 )
447	446	oveq1d	⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 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 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
449	441 445 448	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 ) )
450	409 449	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 ) )
451		iftrue	⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) = if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) )
452	451	oveq1d	⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) )
453		iftrue	⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) = if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) )
454	453	oveq1d	⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) )
455	452 454	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 ) ) )
456	455	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 ) ) )
457		iftrue	⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) = if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) )
458	457	oveq1d	⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 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 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
460	450 456 459	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 ) )
461	186	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 ) )
462	144 282 283	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 ) )
463	161 462	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 )
464	156 144 463 381	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 ) )
465	169 144 463 286	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 ) )
466	464 465	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 ) ) )
467	141 11	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 )
468	156 467 172	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 )
469	169 170 172	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 )
470	468 469	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 ) ) )
471	466 470	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 ) ) )
472	139 65 246	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 )
473	472 144 463 338	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 ) )
474	461 471 473	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 ) )
475		iffalse	⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) = - 𝑎 )
476	475	oveq1d	⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
477		iffalse	⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) = - 𝑏 )
478	477	oveq1d	⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
479	476 478	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 ) ) )
480	479	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 ) ) )
481		iffalse	⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) = - 𝑐 )
482	481	oveq1d	⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 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 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
484	474 480 483	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 ) )
485	460 484	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 ) )
486		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 ) )
487	486	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 ) )
488		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 ) )
489	488	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 ) )
490	487 489	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 ) ) )
491	490	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 ) ) )
492		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 ) )
493	492	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 ) )
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 , 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 ) )
495	485 491 494	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 ) )
496	374 495	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 ) )
497		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 ) ) )
498	497	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 ) )
499		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 ) ) )
500	499	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 ) )
501	498 500	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 ) ) )
502	501	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 ) ) )
503		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 ) ) )
504	503	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 ) )
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 ‘ 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 ) )
506	496 502 505	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 ) )
507	266 506	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 ) )
508	3 6 8 48 85 197 507	3rspcedvdw	⊢ ( ( ( ( ( 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 ) )
509	508	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 ) ) )
510	509	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 ) ) )
511	510	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 ) ) )
512	511	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 ) )
513		nne	⊢ ( ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
514	513	bicomi	⊢ ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
515	514	rexbii	⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
516		rexnal	⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
517	515 516	bitri	⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
518	517	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 ) )
519		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 ) )
520	518 519	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 ) )
521	520	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 ) )
522		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 ) )
523	521 522	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 ) )
524	523	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 ) )
525		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 ) )
526	524 525	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 ) )
527		nne	⊢ ( ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) )
528	527	bicomi	⊢ ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
529	528	rexbii	⊢ ( ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑧 ∈ ℕ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
530		rexnal	⊢ ( ∃ 𝑧 ∈ ℕ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
531	529 530	bitri	⊢ ( ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
532	531	rexbii	⊢ ( ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑦 ∈ ℕ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
533		rexnal	⊢ ( ∃ 𝑦 ∈ ℕ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
534	532 533	bitri	⊢ ( ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
535	534	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 ) )
536		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 ) )
537	535 536	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 ) )
538	537	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 ) )
539		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 ) )
540	538 539	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 ) )
541	512 526 540	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 ) )
542	541	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 ) )
543		dfn2	⊢ ℕ = ( ℕ0 ∖ { 0 } )
544		nn0ssz	⊢ ℕ0 ⊆ ℤ
545		ssdif	⊢ ( ℕ0 ⊆ ℤ → ( ℕ0 ∖ { 0 } ) ⊆ ( ℤ ∖ { 0 } ) )
546	544 545	ax-mp	⊢ ( ℕ0 ∖ { 0 } ) ⊆ ( ℤ ∖ { 0 } )
547	543 546	eqsstri	⊢ ℕ ⊆ ( ℤ ∖ { 0 } )
548		ssel	⊢ ( ℕ ⊆ ( ℤ ∖ { 0 } ) → ( 𝑎 ∈ ℕ → 𝑎 ∈ ( ℤ ∖ { 0 } ) ) )
549		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 ) ) )
550	548 549	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 ) ) ) )
551	550	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 ) ) )
552	547 551	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 ) )
553		oveq1	⊢ ( 𝑎 = 𝑥 → ( 𝑎 ↑ 𝑛 ) = ( 𝑥 ↑ 𝑛 ) )
554	553	oveq1d	⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
555	554	neeq1d	⊢ ( 𝑎 = 𝑥 → ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
556		oveq1	⊢ ( 𝑏 = 𝑦 → ( 𝑏 ↑ 𝑛 ) = ( 𝑦 ↑ 𝑛 ) )
557	556	oveq2d	⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) )
558	557	neeq1d	⊢ ( 𝑏 = 𝑦 → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
559		oveq1	⊢ ( 𝑐 = 𝑧 → ( 𝑐 ↑ 𝑛 ) = ( 𝑧 ↑ 𝑛 ) )
560	559	neeq2d	⊢ ( 𝑐 = 𝑧 → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ) )
561	555 558 560	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 ) )
562	552 561	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 ) )
563	562	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 ) )
564	542 563	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 ) )