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Theorem itgex 22670
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 22523 . 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 13697 . 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 2502 1  |-  S. A B  _d x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    e. wcel 1872   _Vcvv 3022   [_csb 3338   ifcif 3854   class class class wbr 4366    |-> cmpt 4425   ` cfv 5544  (class class class)co 6249   RRcr 9489   0cc0 9490   _ici 9492    x. cmul 9495    <_ cle 9627    / cdiv 10220   3c3 10611   ...cfz 11735   ^cexp 12222   Recre 13104   sum_csu 13695   S.2citg2 22516   S.citg 22518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-nul 4498
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-sn 3942  df-pr 3944  df-uni 4163  df-iota 5508  df-sum 13696  df-itg 22523
This theorem is referenced by:  ditgex  22749  ftc1lem1  22929  itgulm  23305  dmarea  23825  dfarea  23828  areaval  23832  ftc1anc  31932  itgsinexp  37714  wallispilem1  37810  wallispilem2  37811
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