Theorem esumeq2

Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016)

	Ref	Expression
Assertion	esumeq2	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2		mpteq12	⊢ ( ( 𝐴 = 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) )
3	1 2	mpan	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) )
4	3	oveq2d	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) )
5	4	unieqd	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) )
6		df-esum	⊢ Σ* 𝑘 ∈ 𝐴 𝐵 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) )
7		df-esum	⊢ Σ* 𝑘 ∈ 𝐴 𝐶 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) )
8	5 6 7	3eqtr4g	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )