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

Theorem itgex 23343
 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 23198 . 2 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2 sumex 14266 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
31, 2eqeltri 2684 1 𝐴𝐵 d𝑥 ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  ici 9817   · cmul 9820   ≤ cle 9954   / cdiv 10563  3c3 10948  ...cfz 12197  ↑cexp 12722  ℜcre 13685  Σcsu 14264  ∫2citg2 23191  ∫citg 23193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-sum 14265  df-itg 23198 This theorem is referenced by:  ditgex  23422  ftc1lem1  23602  itgulm  23966  dmarea  24484  dfarea  24487  areaval  24491  ftc1anc  32663  itgsinexp  38846  wallispilem1  38958  wallispilem2  38959
 Copyright terms: Public domain W3C validator