Theorem itgex 25699

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 25552	. 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 15597	. 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 395   ∈ wcel 2113  Vcvv 3437  ⦋csb 3846  ifcif 4474   class class class wbr 5093   ↦ cmpt 5174  ‘cfv 6486  (class class class)co 7352  ℝcr 11012  0cc0 11013  ici 11015   · cmul 11018   ≤ cle 11154   / cdiv 11781  3c3 12188  ...cfz 13409  ↑cexp 13970  ℜcre 15006  Σcsu 15595  ∫₂citg2 25545  ∫citg 25547

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 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

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 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-sn 4576  df-pr 4578  df-uni 4859  df-iota 6442  df-sum 15596  df-itg 25552

This theorem is referenced by:  ditgex  25781  ftc1lem1  25970  itgulm  26345  dmarea  26895  dfarea  26898  areaval  26902  ftc1anc  37761  itgsinexp  46077  wallispilem1  46187  wallispilem2  46188