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Theorem itgex 21928
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 21783 . 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 13472 . 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 2551 1  |-  S. A B  _d x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1767   _Vcvv 3113   [_csb 3435   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   ` cfv 5587  (class class class)co 6283   RRcr 9490   0cc0 9491   _ici 9493    x. cmul 9496    <_ cle 9628    / cdiv 10205   3c3 10585   ...cfz 11671   ^cexp 12133   Recre 12892   sum_csu 13470   S.2citg2 21776   S.citg 21778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-uni 4246  df-iota 5550  df-sum 13471  df-itg 21783
This theorem is referenced by:  ditgex  22007  ftc1lem1  22187  itgulm  22553  dmarea  23031  dfarea  23034  areaval  23038  ftc1anc  29691  itgsinexp  31288  wallispilem1  31381  wallispilem2  31382
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