Theorem omeoALTV

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

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

Proof

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