Theorem 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 ∈ Odd ∧ 𝐵 ∈ Even ) → ( 𝐴 − 𝐵 ) ∈ Odd )

Proof

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