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Theorem itgex 19141
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 18995 . 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 12176 . 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 2366 1  |-  S. A B  _d x  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1696   _Vcvv 2801   [_csb 3094   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   _ici 8755    x. cmul 8758    <_ cle 8884    / cdiv 9439   3c3 9812   ...cfz 10798   ^cexp 11120   Recre 11598   sum_csu 12174   S.2citg2 18987   S.citg 18989
This theorem is referenced by:  ditgex  19218  ftc1lem1  19398  itgulm  19800  dmarea  20268  dfarea  20271  areaval  20275  itgsinexp  27852  wallispilem1  27917  wallispilem2  27918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-sum 12175  df-itg 18995
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