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

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

Metamath Proof Explorer