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Theorem itgex 24381
 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 24234 . 2 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2 sumex 15038 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
31, 2eqeltri 2886 1 𝐴𝐵 d𝑥 ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∈ wcel 2111  Vcvv 3441  ⦋csb 3828  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  ℝcr 10527  0cc0 10528  ici 10530   · cmul 10533   ≤ cle 10667   / cdiv 11288  3c3 11683  ...cfz 12887  ↑cexp 13427  ℜcre 14450  Σcsu 15036  ∫2citg2 24227  ∫citg 24229 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4801  df-iota 6283  df-sum 15037  df-itg 24234 This theorem is referenced by:  ditgex  24462  ftc1lem1  24645  itgulm  25010  dmarea  25550  dfarea  25553  areaval  25557  ftc1anc  35154  itgsinexp  42612  wallispilem1  42722  wallispilem2  42723
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