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Metamath Proof Explorer

Theorem omoeALTV

Description: The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014) (Revised by AV, 20-Jun-2020)

Ref Expression
Assertion omoeALTV 
|- ( ( A e. Odd /\ B e. Odd ) -> ( A - B ) e. Even )

Proof

Step Hyp Ref Expression
1 oddz 
 |-  ( A e. Odd -> A e. ZZ )
2 1 zcnd 
 |-  ( A e. Odd -> A e. CC )
3 oddz 
 |-  ( B e. Odd -> B e. ZZ )
4 3 zcnd 
 |-  ( B e. Odd -> B e. CC )
5 negsub 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
6 2 4 5 syl2an 
 |-  ( ( A e. Odd /\ B e. Odd ) -> ( A + -u B ) = ( A - B ) )
7 onego 
 |-  ( B e. Odd -> -u B e. Odd )
8 opoeALTV 
 |-  ( ( A e. Odd /\ -u B e. Odd ) -> ( A + -u B ) e. Even )
9 7 8 sylan2 
 |-  ( ( A e. Odd /\ B e. Odd ) -> ( A + -u B ) e. Even )
10 6 9 eqeltrrd 
 |-  ( ( A e. Odd /\ B e. Odd ) -> ( A - B ) e. Even )