Theorem zrtdvds

Description: A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023)

	Ref	Expression
Assertion	zrtdvds	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> ( A ^c ( 1 / N ) ) || A )

Proof

Step	Hyp	Ref	Expression
1		nnz	|- ( ( A ^c ( 1 / N ) ) e. NN -> ( A ^c ( 1 / N ) ) e. ZZ )
2	1	3ad2ant3	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> ( A ^c ( 1 / N ) ) e. ZZ )
3		simp2	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> N e. NN )
4		iddvdsexp	|- ( ( ( A ^c ( 1 / N ) ) e. ZZ /\ N e. NN ) -> ( A ^c ( 1 / N ) ) || ( ( A ^c ( 1 / N ) ) ^ N ) )
5	2 3 4	syl2anc	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> ( A ^c ( 1 / N ) ) || ( ( A ^c ( 1 / N ) ) ^ N ) )
6		zcn	|- ( A e. ZZ -> A e. CC )
7	6	3ad2ant1	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> A e. CC )
8		cxproot	|- ( ( A e. CC /\ N e. NN ) -> ( ( A ^c ( 1 / N ) ) ^ N ) = A )
9	7 3 8	syl2anc	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> ( ( A ^c ( 1 / N ) ) ^ N ) = A )
10	5 9	breqtrd	|- ( ( A e. ZZ /\ N e. NN /\ ( A ^c ( 1 / N ) ) e. NN ) -> ( A ^c ( 1 / N ) ) || A )