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Theorem itgex 24064
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 23917 . 2 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2 sumex 14895 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
31, 2eqeltri 2856 1 𝐴𝐵 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 387  wcel 2048  Vcvv 3409  csb 3782  ifcif 4344   class class class wbr 4923  cmpt 5002  cfv 6182  (class class class)co 6970  cr 10326  0cc0 10327  ici 10329   · cmul 10332  cle 10467   / cdiv 11090  3c3 11489  ...cfz 12701  cexp 13237  cre 14307  Σcsu 14893  2citg2 23910  citg 23912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-nul 5061
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-sn 4436  df-pr 4438  df-uni 4707  df-iota 6146  df-sum 14894  df-itg 23917
This theorem is referenced by:  ditgex  24143  ftc1lem1  24325  itgulm  24689  dmarea  25227  dfarea  25230  areaval  25234  ftc1anc  34364  itgsinexp  41616  wallispilem1  41727  wallispilem2  41728
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