Theorem itgex 25801

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 25654	. 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 15687	. 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 2848	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 398   ∈ wcel 2132  Vcvv 3444  ⦋csb 3843  ifcif 4470   class class class wbr 5090   ↦ cmpt 5171  ‘cfv 6506  (class class class)co 7381  ℝcr 11058  0cc0 11059  ici 11061   · cmul 11064   ≤ cle 11203   / cdiv 11830  3c3 12259  ...cfz 13498  ↑cexp 14060  ℜcre 15096  Σcsu 15685  ∫₂citg2 25647  ∫citg 25649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-nul 5246

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-v 3446  df-dif 3898  df-un 3900  df-ss 3912  df-nul 4277  df-sn 4573  df-pr 4575  df-uni 4856  df-iota 6462  df-sum 15686  df-itg 25654

This theorem is referenced by:  ditgex  25883  ftc1lem1  26066  itgulm  26437  dmarea  26988  dfarea  26991  areaval  26995  ftc1anc  38138  itgsinexp  46467  wallispilem1  46577  wallispilem2  46578