Theorem m1expcl

Description: Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015)

	Ref	Expression
Assertion	m1expcl	|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		neg1z	|- -u 1 e. ZZ
2		1z	|- 1 e. ZZ
3		prssi	|- ( ( -u 1 e. ZZ /\ 1 e. ZZ ) -> { -u 1 , 1 } C_ ZZ )
4	1 2 3	mp2an	|- { -u 1 , 1 } C_ ZZ
5		m1expcl2	|- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
6	4 5	sseldi	|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )