Theorem sumeq2

Description: Equality theorem for sum. (Contributed by NM, 11-Dec-2005) (Revised by Mario Carneiro, 13-Jul-2013)

	Ref	Expression
Assertion	sumeq2	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐵 = 𝐶 → ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) )
2	1	ralimi	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) )
3		sumeq2ii	⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )
4	2 3	syl	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )