Theorem esumex 34010

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 34009	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		ovex 7464	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
3	2	uniex 7760	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
4	1, 3	eqeltri 2835	1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478  ∪ cuni 4912   ↦ cmpt 5231  (class class class)co 7431  0cc0 11153  +∞cpnf 11290  [,]cicc 13387   ↾_s cress 17274  ℝ^*_𝑠cxrs 17547   tsums ctsu 24150  Σ^*cesum 34008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571  df-ov 7434  df-esum 34009

This theorem is referenced by:  esumcvg  34067  esumgect  34071  omssubaddlem  34281  omssubadd  34282