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Theorem nfitg1 22362
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 22214 . 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 2562 . . 3  |-  F/_ x
( 0 ... 3
)
3 nfcv 2562 . . . 4  |-  F/_ x
( _i ^ k
)
4 nfcv 2562 . . . 4  |-  F/_ x  x.
5 nfcv 2562 . . . . 5  |-  F/_ x S.2
6 nfmpt1 4481 . . . . 5  |-  F/_ x
( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) )
75, 6nffv 5810 . . . 4  |-  F/_ x
( S.2 `  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  z ]_ if ( ( x  e.  A  /\  0  <_ 
z ) ,  z ,  0 ) ) )
83, 4, 7nfov 6258 . . 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 13567 . 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 2560 1  |-  F/_ x S. A B  _d x
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    e. wcel 1840   F/_wnfc 2548   [_csb 3370   ifcif 3882   class class class wbr 4392    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232   RRcr 9439   0cc0 9440   _ici 9442    x. cmul 9445    <_ cle 9577    / cdiv 10165   3c3 10545   ...cfz 11641   ^cexp 12118   Recre 12984   sum_csu 13562   S.2citg2 22207   S.citg 22209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-recs 6997  df-rdg 7031  df-seq 12060  df-sum 13563  df-itg 22214
This theorem is referenced by: (None)
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