Theorem evensumodd

Description: If every term in a sum with an even number of terms is odd, then the sum is even. (Contributed by AV, 14-Aug-2021)

	Ref	Expression
Hypotheses	evensumodd.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	evensumodd.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℤ )
	evensumodd.o	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ¬ 2 ∥ 𝐵 )
	evensumodd.e	⊢ ( 𝜑 → 2 ∥ ( ♯ ‘ 𝐴 ) )
Assertion	evensumodd	⊢ ( 𝜑 → 2 ∥ Σ 𝑘 ∈ 𝐴 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		evensumodd.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		evensumodd.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℤ )
3		evensumodd.o	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ¬ 2 ∥ 𝐵 )
4		evensumodd.e	⊢ ( 𝜑 → 2 ∥ ( ♯ ‘ 𝐴 ) )
5	1 2 3	sumodd	⊢ ( 𝜑 → ( 2 ∥ ( ♯ ‘ 𝐴 ) ↔ 2 ∥ Σ 𝑘 ∈ 𝐴 𝐵 ) )
6	4 5	mpbid	⊢ ( 𝜑 → 2 ∥ Σ 𝑘 ∈ 𝐴 𝐵 )