Theorem itgex 25287

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 25139	. 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 15633	. 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 2829	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 396   ∈ wcel 2106  Vcvv 3474  ⦋csb 3893  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6543  (class class class)co 7408  ℝcr 11108  0cc0 11109  ici 11111   · cmul 11114   ≤ cle 11248   / cdiv 11870  3c3 12267  ...cfz 13483  ↑cexp 14026  ℜcre 15043  Σcsu 15631  ∫₂citg2 25132  ∫citg 25134

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-sum 15632  df-itg 25139

This theorem is referenced by:  ditgex  25368  ftc1lem1  25551  itgulm  25919  dmarea  26459  dfarea  26462  areaval  26466  ftc1anc  36564  itgsinexp  44661  wallispilem1  44771  wallispilem2  44772