Theorem esumex 31897

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 31896	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		ovex 7288	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
3	2	uniex 7572	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
4	1, 3	eqeltri 2835	1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3422  ∪ cuni 4836   ↦ cmpt 5153  (class class class)co 7255  0cc0 10802  +∞cpnf 10937  [,]cicc 13011   ↾_s cress 16867  ℝ^*_𝑠cxrs 17128   tsums ctsu 23185  Σ^*cesum 31895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6376  df-fv 6426  df-ov 7258  df-esum 31896

This theorem is referenced by:  esumcvg  31954  esumgect  31958  omssubaddlem  32166  omssubadd  32167