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Theorem itgex 19663
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 19517 . 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 12482 . 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 2507 1  |-  S. A B  _d x  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 360    e. wcel 1726   _Vcvv 2957   [_csb 3252   ifcif 3740   class class class wbr 4213    e. cmpt 4267   ` cfv 5455  (class class class)co 6082   RRcr 8990   0cc0 8991   _ici 8993    x. cmul 8996    <_ cle 9122    / cdiv 9678   3c3 10051   ...cfz 11044   ^cexp 11383   Recre 11903   sum_csu 12480   S.2citg2 19509   S.citg 19511
This theorem is referenced by:  ditgex  19740  ftc1lem1  19920  itgulm  20325  dmarea  20797  dfarea  20800  areaval  20804  ftc1anc  26289  itgsinexp  27726  wallispilem1  27791  wallispilem2  27792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-sn 3821  df-pr 3822  df-uni 4017  df-iota 5419  df-sum 12481  df-itg 19517
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