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Theorem nfitg1 19343
Description: Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
nfitg1  |-  F/_ x S. A B  _d x

Proof of Theorem nfitg1
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-itg 19194 . 2  |-  S. A B  _d x  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) ) ) )
2 nfcv 2502 . . 3  |-  F/_ x
( 0 ... 3
)
3 nfcv 2502 . . . 4  |-  F/_ x
( _i ^ k
)
4 nfcv 2502 . . . 4  |-  F/_ x  x.
5 nfcv 2502 . . . . 5  |-  F/_ x S.2
6 nfmpt1 4211 . . . . 5  |-  F/_ x
( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) )
75, 6nffv 5639 . . . 4  |-  F/_ x
( S.2 `  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) ) )
83, 4, 7nfov 6004 . . 3  |-  F/_ x
( ( _i ^
k )  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  (
_i ^ k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_  z ) ,  z ,  0 ) ) ) )
92, 8nfsum 12372 . 2  |-  F/_ x sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) ) ) )
101, 9nfcxfr 2499 1  |-  F/_ x S. A B  _d x
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1715   F/_wnfc 2489   [_csb 3167   ifcif 3654   class class class wbr 4125    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   RRcr 8883   0cc0 8884   _ici 8886    x. cmul 8889    <_ cle 9015    / cdiv 9570   3c3 9943   ...cfz 10935   ^cexp 11269   Recre 11789   sum_csu 12366   S.2citg2 19186   S.citg 19188
This theorem is referenced by:  itgabsnc  25777  ftc1cnnclem  25781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-recs 6530  df-rdg 6565  df-seq 11211  df-sum 12367  df-itg 19194
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