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	$⊢ (A \in Odd \land B \in Even) \to A - B \in Odd$

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

Metamath Proof Explorer