Theorem esumex 32751

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 32750	. 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		ovex 7410	. . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
3	2	uniex 7698	. 2 ⊢ ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
4	1, 3	eqeltri 2828	1 ⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3459  ∪ cuni 4885   ↦ cmpt 5208  (class class class)co 7377  0cc0 11075  +∞cpnf 11210  [,]cicc 13292   ↾_s cress 17138  ℝ^*_𝑠cxrs 17411   tsums ctsu 23529  Σ^*cesum 32749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-sn 4607  df-pr 4609  df-uni 4886  df-iota 6468  df-fv 6524  df-ov 7380  df-esum 32750

This theorem is referenced by:  esumcvg  32808  esumgect  32812  omssubaddlem  33022  omssubadd  33023