epoo

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

		Ref	Expression
	Assertion	epoo	$⊢ (A \in Even \land B \in Odd) \to A + B \in Odd$

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		addcom	$⊢ (A \in ℂ \land B \in ℂ) \to A + B = B + A$
6	2 4 5	syl2an	$⊢ (A \in Even \land B \in Odd) \to A + B = B + A$
7		opeoALTV	$⊢ (B \in Odd \land A \in Even) \to B + A \in Odd$
8	7	ancoms	$⊢ (A \in Even \land B \in Odd) \to B + A \in Odd$
9	6 8	eqeltrd	$⊢ (A \in Even \land B \in Odd) \to A + B \in Odd$

Metamath Proof Explorer