Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumex Structured version   Visualization version   GIF version

Theorem esumex 31187
Description: An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Assertion
Ref Expression
esumex Σ*𝑘𝐴𝐵 ∈ V

Proof of Theorem esumex
StepHypRef Expression
1 df-esum 31186 . 2 Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))
2 ovex 7178 . . 3 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) ∈ V
32uniex 7454 . 2 ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)) ∈ V
41, 3eqeltri 2906 1 Σ*𝑘𝐴𝐵 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  Vcvv 3492   cuni 4830  cmpt 5137  (class class class)co 7145  0cc0 10525  +∞cpnf 10660  [,]cicc 12729  s cress 16472  *𝑠cxrs 16761   tsums ctsu 22661  Σ*cesum 31185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307  df-fv 6356  df-ov 7148  df-esum 31186
This theorem is referenced by:  esumcvg  31244  esumgect  31248  omssubaddlem  31456  omssubadd  31457
  Copyright terms: Public domain W3C validator