Theorem esumex 30410

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 30409	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		ovex 6900	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
3	2	uniex 7177	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
4	1, 3	eqeltri 2877	1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V

Syntax hints:   ∈ wcel 2155  Vcvv 3387  ∪ cuni 4623   ↦ cmpt 4916  (class class class)co 6868  0cc0 10215  +∞cpnf 10350  [,]cicc 12390   ↾_s cress 16063  ℝ^*_𝑠cxrs 16359   tsums ctsu 22136  Σ^*cesum 30408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-un 7173

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-sn 4365  df-pr 4367  df-uni 4624  df-iota 6058  df-fv 6103  df-ov 6871  df-esum 30409

This theorem is referenced by:  esumcvg  30467  esumgect  30471  omssubaddlem  30680  omssubadd  30681