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Theorem itgex 19125
Description: An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
itgex  |-  S. A B  _d x  e.  _V

Proof of Theorem itgex
Dummy variables  k 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-itg 18979 . 2  |-  S. A B  _d x  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) ) )
2 sumex 12160 . 2  |-  sum_ k  e.  ( 0 ... 3
) ( ( _i
^ k )  x.  ( S.2 `  (
x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) ) )  e.  _V
31, 2eqeltri 2353 1  |-  S. A B  _d x  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1684   _Vcvv 2788   [_csb 3081   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   _ici 8739    x. cmul 8742    <_ cle 8868    / cdiv 9423   3c3 9796   ...cfz 10782   ^cexp 11104   Recre 11582   sum_csu 12158   S.2citg2 18971   S.citg 18973
This theorem is referenced by:  ditgex  19202  ftc1lem1  19382  itgulm  19784  dmarea  20252  dfarea  20255  areaval  20259  itgsinexp  27749  wallispilem1  27814  wallispilem2  27815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-sum 12159  df-itg 18979
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