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Theorem itgex 25620
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 25472 . 2 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2 sumex 15641 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
31, 2eqeltri 2828 1 𝐴𝐵 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2105  Vcvv 3473  csb 3893  ifcif 4528   class class class wbr 5148  cmpt 5231  cfv 6543  (class class class)co 7412  cr 11115  0cc0 11116  ici 11118   · cmul 11121  cle 11256   / cdiv 11878  3c3 12275  ...cfz 13491  cexp 14034  cre 15051  Σcsu 15639  2citg2 25465  citg 25467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-sum 15640  df-itg 25472
This theorem is referenced by:  ditgex  25701  ftc1lem1  25890  itgulm  26259  dmarea  26803  dfarea  26806  areaval  26810  ftc1anc  37033  itgsinexp  45130  wallispilem1  45240  wallispilem2  45241
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