Theorem itgex 25696

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.

Step	Hyp	Ref	Expression
1		df-itg 25549	. 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2		sumex 15592	. 2 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
3	1, 2	eqeltri 2827	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 395   ∈ wcel 2111  Vcvv 3436  ⦋csb 3850  ifcif 4475   class class class wbr 5091   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346  ℝcr 11002  0cc0 11003  ici 11005   · cmul 11008   ≤ cle 11144   / cdiv 11771  3c3 12178  ...cfz 13404  ↑cexp 13965  ℜcre 15001  Σcsu 15590  ∫₂citg2 25542  ∫citg 25544

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-sn 4577  df-pr 4579  df-uni 4860  df-iota 6437  df-sum 15591  df-itg 25549

This theorem is referenced by:  ditgex  25778  ftc1lem1  25967  itgulm  26342  dmarea  26892  dfarea  26895  areaval  26899  ftc1anc  37740  itgsinexp  45992  wallispilem1  46102  wallispilem2  46103