Theorem itgex 25762

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 25615	. 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 15648	. 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 2836	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 396   ∈ wcel 2119  Vcvv 3432  ⦋csb 3838  ifcif 4461   class class class wbr 5079   ↦ cmpt 5160  ‘cfv 6492  (class class class)co 7363  ℝcr 11035  0cc0 11036  ici 11038   · cmul 11041   ≤ cle 11178   / cdiv 11805  3c3 12235  ...cfz 13459  ↑cexp 14021  ℜcre 15057  Σcsu 15646  ∫₂citg2 25608  ∫citg 25610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-sn 4563  df-pr 4565  df-uni 4846  df-iota 6448  df-sum 15647  df-itg 25615

This theorem is referenced by:  ditgex  25844  ftc1lem1  26027  itgulm  26398  dmarea  26946  dfarea  26949  areaval  26953  ftc1anc  38075  itgsinexp  46405  wallispilem1  46515  wallispilem2  46516