Theorem nnennex

Description: For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023)

	Ref	Expression
Assertion	nnennex	|- ( ( N e. NN /\ ( N / 2 ) e. NN ) -> E. m e. NN N = ( 2 x. m ) )

Proof

Step	Hyp	Ref	Expression
1		simpr	|- ( ( N e. NN /\ ( N / 2 ) e. NN ) -> ( N / 2 ) e. NN )
2		oveq2	|- ( m = ( N / 2 ) -> ( 2 x. m ) = ( 2 x. ( N / 2 ) ) )
3	2	eqeq2d	|- ( m = ( N / 2 ) -> ( N = ( 2 x. m ) <-> N = ( 2 x. ( N / 2 ) ) ) )
4	3	adantl	|- ( ( ( N e. NN /\ ( N / 2 ) e. NN ) /\ m = ( N / 2 ) ) -> ( N = ( 2 x. m ) <-> N = ( 2 x. ( N / 2 ) ) ) )
5		nncn	|- ( N e. NN -> N e. CC )
6		2cnd	|- ( N e. NN -> 2 e. CC )
7		2ne0	|- 2 =/= 0
8	7	a1i	|- ( N e. NN -> 2 =/= 0 )
9		divcan2	|- ( ( N e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( 2 x. ( N / 2 ) ) = N )
10	9	eqcomd	|- ( ( N e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> N = ( 2 x. ( N / 2 ) ) )
11	5 6 8 10	syl3anc	|- ( N e. NN -> N = ( 2 x. ( N / 2 ) ) )
12	11	adantr	|- ( ( N e. NN /\ ( N / 2 ) e. NN ) -> N = ( 2 x. ( N / 2 ) ) )
13	1 4 12	rspcedvd	|- ( ( N e. NN /\ ( N / 2 ) e. NN ) -> E. m e. NN N = ( 2 x. m ) )