Theorem itgex 25620

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 25472	. 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 15641	. 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 2828	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 395   ∈ wcel 2105  Vcvv 3473  ⦋csb 3893  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6543  (class class class)co 7412  ℝcr 11115  0cc0 11116  ici 11118   · cmul 11121   ≤ cle 11256   / cdiv 11878  3c3 12275  ...cfz 13491  ↑cexp 14034  ℜcre 15051  Σcsu 15639  ∫₂citg2 25465  ∫citg 25467

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  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 15640  df-itg 25472

This theorem is referenced by:  ditgex  25701  ftc1lem1  25890  itgulm  26259  dmarea  26803  dfarea  26806  areaval  26810  ftc1anc  37033  itgsinexp  45130  wallispilem1  45240  wallispilem2  45241