Theorem itgex 24963

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 24815	. 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 15427	. 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 2830	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 395   ∈ wcel 2101  Vcvv 3434  ⦋csb 3834  ifcif 4462   class class class wbr 5077   ↦ cmpt 5160  ‘cfv 6447  (class class class)co 7295  ℝcr 10898  0cc0 10899  ici 10901   · cmul 10904   ≤ cle 11038   / cdiv 11660  3c3 12057  ...cfz 13267  ↑cexp 13810  ℜcre 14836  Σcsu 15425  ∫₂citg2 24808  ∫citg 24810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2939  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-sn 4565  df-pr 4567  df-uni 4842  df-iota 6399  df-sum 15426  df-itg 24815

This theorem is referenced by:  ditgex  25044  ftc1lem1  25227  itgulm  25595  dmarea  26135  dfarea  26138  areaval  26142  ftc1anc  35886  itgsinexp  43531  wallispilem1  43641  wallispilem2  43642