emoo

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

		Ref	Expression
	Assertion	emoo	$⊢ (A \in Even \land B \in Odd) \to A - B \in Odd$

Step	Hyp	Ref	Expression
1		evenz	$⊢ A \in Even \to A \in ℤ$
2	1	zcnd	$⊢ A \in Even \to A \in ℂ$
3		oddz	$⊢ B \in Odd \to B \in ℤ$
4	3	zcnd	$⊢ B \in Odd \to B \in ℂ$
5		negsub	$⊢ (A \in ℂ \land B \in ℂ) \to A + (- B) = A - B$
6	2 4 5	syl2an	$⊢ (A \in Even \land B \in Odd) \to A + (- B) = A - B$
7		onego	$⊢ B \in Odd \to - B \in Odd$
8		epoo	$⊢ (A \in Even \land - B \in Odd) \to A + (- B) \in Odd$
9	7 8	sylan2	$⊢ (A \in Even \land B \in Odd) \to A + (- B) \in Odd$
10	6 9	eqeltrrd	$⊢ (A \in Even \land B \in Odd) \to A - B \in Odd$

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