Theorem leibpilem1

Description: Lemma for leibpi . (Contributed by Mario Carneiro, 7-Apr-2015) (Proof shortened by Steven Nguyen, 23-Mar-2023)

	Ref	Expression
Assertion	leibpilem1	|- ( ( N e. NN0 /\ ( -. N = 0 /\ -. 2 || N ) ) -> ( N e. NN /\ ( ( N - 1 ) / 2 ) e. NN0 ) )

Proof

Step	Hyp	Ref	Expression
1		nn0onn	|- ( ( N e. NN0 /\ -. 2 || N ) -> N e. NN )
2		nn0oddm1d2	|- ( N e. NN0 -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. NN0 ) )
3	2	biimpa	|- ( ( N e. NN0 /\ -. 2 || N ) -> ( ( N - 1 ) / 2 ) e. NN0 )
4	1 3	jca	|- ( ( N e. NN0 /\ -. 2 || N ) -> ( N e. NN /\ ( ( N - 1 ) / 2 ) e. NN0 ) )
5	4	adantrl	|- ( ( N e. NN0 /\ ( -. N = 0 /\ -. 2 || N ) ) -> ( N e. NN /\ ( ( N - 1 ) / 2 ) e. NN0 ) )