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Metamath Proof Explorer

Theorem nnoddn2prm

Description: A prime not equal to 2 is an odd positive integer. (Contributed by AV, 28-Jun-2021)

Ref Expression
Assertion nnoddn2prm 
|- ( N e. ( Prime \ { 2 } ) -> ( N e. NN /\ -. 2 || N ) )

Proof

Step Hyp Ref Expression
1 eldifi 
 |-  ( N e. ( Prime \ { 2 } ) -> N e. Prime )
2 prmnn 
 |-  ( N e. Prime -> N e. NN )
3 1 2 syl 
 |-  ( N e. ( Prime \ { 2 } ) -> N e. NN )
4 oddprm 
 |-  ( N e. ( Prime \ { 2 } ) -> ( ( N - 1 ) / 2 ) e. NN )
5 nnz 
 |-  ( ( ( N - 1 ) / 2 ) e. NN -> ( ( N - 1 ) / 2 ) e. ZZ )
6 nnz 
 |-  ( N e. NN -> N e. ZZ )
7 oddm1d2 
 |-  ( N e. ZZ -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
8 6 7 syl 
 |-  ( N e. NN -> ( -. 2 || N <-> ( ( N - 1 ) / 2 ) e. ZZ ) )
9 5 8 syl5ibrcom 
 |-  ( ( ( N - 1 ) / 2 ) e. NN -> ( N e. NN -> -. 2 || N ) )
10 4 9 syl 
 |-  ( N e. ( Prime \ { 2 } ) -> ( N e. NN -> -. 2 || N ) )
11 3 10 jcai 
 |-  ( N e. ( Prime \ { 2 } ) -> ( N e. NN /\ -. 2 || N ) )