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Metamath Proof Explorer

Theorem sbgoldbaltlem2

Description: Lemma 2 for sbgoldbalt : If an even number greater than 4 is the sum of two primes, the primes must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020)

Ref Expression
Assertion sbgoldbaltlem2 
|- ( ( P e. Prime /\ Q e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( P + Q ) ) -> ( P e. Odd /\ Q e. Odd ) ) )

Proof

Step Hyp Ref Expression
1 prmz 
 |-  ( P e. Prime -> P e. ZZ )
2 1 zcnd 
 |-  ( P e. Prime -> P e. CC )
3 prmz 
 |-  ( Q e. Prime -> Q e. ZZ )
4 3 zcnd 
 |-  ( Q e. Prime -> Q e. CC )
5 addcom 
 |-  ( ( P e. CC /\ Q e. CC ) -> ( P + Q ) = ( Q + P ) )
6 2 4 5 syl2anr 
 |-  ( ( Q e. Prime /\ P e. Prime ) -> ( P + Q ) = ( Q + P ) )
7 6 eqeq2d 
 |-  ( ( Q e. Prime /\ P e. Prime ) -> ( N = ( P + Q ) <-> N = ( Q + P ) ) )
8 7 3anbi3d 
 |-  ( ( Q e. Prime /\ P e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( P + Q ) ) <-> ( N e. Even /\ 4 < N /\ N = ( Q + P ) ) ) )
9 sbgoldbaltlem1 
 |-  ( ( Q e. Prime /\ P e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( Q + P ) ) -> P e. Odd ) )
10 8 9 sylbid 
 |-  ( ( Q e. Prime /\ P e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( P + Q ) ) -> P e. Odd ) )
11 10 ancoms 
 |-  ( ( P e. Prime /\ Q e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( P + Q ) ) -> P e. Odd ) )
12 sbgoldbaltlem1 
 |-  ( ( P e. Prime /\ Q e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( P + Q ) ) -> Q e. Odd ) )
13 11 12 jcad 
 |-  ( ( P e. Prime /\ Q e. Prime ) -> ( ( N e. Even /\ 4 < N /\ N = ( P + Q ) ) -> ( P e. Odd /\ Q e. Odd ) ) )