Theorem sumeq12i

Description: Equality inference for sum. (Contributed by FL, 10-Dec-2006)

	Ref	Expression
Hypotheses	sumeq12i.1	⊢ 𝐴 = 𝐵
	sumeq12i.2	⊢ ( 𝑘 ∈ 𝐴 → 𝐶 = 𝐷 )
Assertion	sumeq12i	⊢ Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐷

Proof

Step	Hyp	Ref	Expression
1		sumeq12i.1	⊢ 𝐴 = 𝐵
2		sumeq12i.2	⊢ ( 𝑘 ∈ 𝐴 → 𝐶 = 𝐷 )
3	2	sumeq2i	⊢ Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐴 𝐷
4	1	sumeq1i	⊢ Σ 𝑘 ∈ 𝐴 𝐷 = Σ 𝑘 ∈ 𝐵 𝐷
5	3 4	eqtri	⊢ Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐷