Theorem nfesum1

Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017)

	Ref	Expression
Hypothesis	nfesum1.1	⊢ Ⅎ 𝑘 𝐴
Assertion	nfesum1	⊢ Ⅎ 𝑘 Σ* 𝑘 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfesum1.1	⊢ Ⅎ 𝑘 𝐴
2		df-esum	⊢ Σ* 𝑘 ∈ 𝐴 𝐵 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) )
3		nfcv	⊢ Ⅎ 𝑘 ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) )
4		nfcv	⊢ Ⅎ 𝑘 tsums
5		nfmpt1	⊢ Ⅎ 𝑘 ( 𝑘 ∈ 𝐴 ↦ 𝐵 )
6	3 4 5	nfov	⊢ Ⅎ 𝑘 ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) )
7	6	nfuni	⊢ Ⅎ 𝑘 ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) )
8	2 7	nfcxfr	⊢ Ⅎ 𝑘 Σ* 𝑘 ∈ 𝐴 𝐵