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Theorem nfitg1 21910
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 21762 . 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 2624 . . 3  |-  F/_ x
( 0 ... 3
)
3 nfcv 2624 . . . 4  |-  F/_ x
( _i ^ k
)
4 nfcv 2624 . . . 4  |-  F/_ x  x.
5 nfcv 2624 . . . . 5  |-  F/_ x S.2
6 nfmpt1 4531 . . . . 5  |-  F/_ x
( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) )
75, 6nffv 5866 . . . 4  |-  F/_ x
( S.2 `  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) ) )
83, 4, 7nfov 6300 . . 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 13464 . 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 2622 1  |-  F/_ x S. A B  _d x
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1762   F/_wnfc 2610   [_csb 3430   ifcif 3934   class class class wbr 4442    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   RRcr 9482   0cc0 9483   _ici 9485    x. cmul 9488    <_ cle 9620    / cdiv 10197   3c3 10577   ...cfz 11663   ^cexp 12124   Recre 12882   sum_csu 13459   S.2citg2 21755   S.citg 21757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-recs 7034  df-rdg 7068  df-seq 12066  df-sum 13460  df-itg 21762
This theorem is referenced by: (None)
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