Theorem esumex 34327

Description: An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)

Assertion

Ref	Expression
esumex	⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V

Proof of Theorem esumex

Step	Hyp	Ref	Expression
1		df-esum 34326	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		ovex 7430	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
3	2	uniex 7725	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
4	1, 3	eqeltri 2859	1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V

Syntax hints:   ∈ wcel 2143  Vcvv 3455  ∪ cuni 4866   ↦ cmpt 5182  (class class class)co 7397  0cc0 11074  +∞cpnf 11214  [,]cicc 13353   ↾_s cress 17267  ℝ^*_𝑠cxrs 17531   tsums ctsu 24187  Σ^*cesum 34325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-sn 4584  df-pr 4586  df-uni 4867  df-iota 6478  df-fv 6530  df-ov 7400  df-esum 34326

This theorem is referenced by:  esumcvg  34384  esumgect  34388  omssubaddlem  34597  omssubadd  34598