MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itgex Structured version   Visualization version   GIF version

Theorem itgex 25820
Description: An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
itgex 𝐴𝐵 d𝑥 ∈ V

Proof of Theorem itgex
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-itg 25672 . 2 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2 sumex 15721 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
31, 2eqeltri 2835 1 𝐴𝐵 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  Vcvv 3478  csb 3908  ifcif 4531   class class class wbr 5148  cmpt 5231  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  ici 11155   · cmul 11158  cle 11294   / cdiv 11918  3c3 12320  ...cfz 13544  cexp 14099  cre 15133  Σcsu 15719  2citg2 25665  citg 25667
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-nul 5312
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-sum 15720  df-itg 25672
This theorem is referenced by:  ditgex  25902  ftc1lem1  26091  itgulm  26466  dmarea  27015  dfarea  27018  areaval  27022  ftc1anc  37688  itgsinexp  45911  wallispilem1  46021  wallispilem2  46022
  Copyright terms: Public domain W3C validator