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

Theorem itgex 23287
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 23142 . 2 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2 sumex 14214 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
31, 2eqeltri 2683 1 𝐴𝐵 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 382  wcel 1976  Vcvv 3172  csb 3498  ifcif 4035   class class class wbr 4577  cmpt 4637  cfv 5789  (class class class)co 6526  cr 9791  0cc0 9792  ici 9794   · cmul 9797  cle 9931   / cdiv 10535  3c3 10920  ...cfz 12154  cexp 12679  cre 13633  Σcsu 14212  2citg2 23135  citg 23137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4711
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-sn 4125  df-pr 4127  df-uni 4367  df-iota 5753  df-sum 14213  df-itg 23142
This theorem is referenced by:  ditgex  23366  ftc1lem1  23546  itgulm  23910  dmarea  24428  dfarea  24431  areaval  24435  ftc1anc  32446  itgsinexp  38629  wallispilem1  38741  wallispilem2  38742
  Copyright terms: Public domain W3C validator