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Theorem nfitg1 19626
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 19477 . 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 2548 . . 3  |-  F/_ x
( 0 ... 3
)
3 nfcv 2548 . . . 4  |-  F/_ x
( _i ^ k
)
4 nfcv 2548 . . . 4  |-  F/_ x  x.
5 nfcv 2548 . . . . 5  |-  F/_ x S.2
6 nfmpt1 4266 . . . . 5  |-  F/_ x
( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) )
75, 6nffv 5702 . . . 4  |-  F/_ x
( S.2 `  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) ) )
83, 4, 7nfov 6071 . . 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 12448 . 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 2545 1  |-  F/_ x S. A B  _d x
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1721   F/_wnfc 2535   [_csb 3219   ifcif 3707   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   _ici 8956    x. cmul 8959    <_ cle 9085    / cdiv 9641   3c3 10014   ...cfz 11007   ^cexp 11345   Recre 11865   sum_csu 12442   S.2citg2 19469   S.citg 19471
This theorem is referenced by:  itgabsnc  26181  ftc1cnnclem  26185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-seq 11287  df-sum 12443  df-itg 19477
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