Theorem emoo

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

	Ref	Expression
Assertion	emoo	⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → ( 𝐴 − 𝐵 ) ∈ Odd )

Proof

Step	Hyp	Ref	Expression
1		evenz	⊢ ( 𝐴 ∈ Even → 𝐴 ∈ ℤ )
2	1	zcnd	⊢ ( 𝐴 ∈ Even → 𝐴 ∈ ℂ )
3		oddz	⊢ ( 𝐵 ∈ Odd → 𝐵 ∈ ℤ )
4	3	zcnd	⊢ ( 𝐵 ∈ Odd → 𝐵 ∈ ℂ )
5		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
6	2 4 5	syl2an	⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
7		onego	⊢ ( 𝐵 ∈ Odd → - 𝐵 ∈ Odd )
8		epoo	⊢ ( ( 𝐴 ∈ Even ∧ - 𝐵 ∈ Odd ) → ( 𝐴 + - 𝐵 ) ∈ Odd )
9	7 8	sylan2	⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → ( 𝐴 + - 𝐵 ) ∈ Odd )
10	6 9	eqeltrrd	⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → ( 𝐴 − 𝐵 ) ∈ Odd )