Theorem emoo

Description: The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020)

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

Proof

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