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Theorem itgvallem 19139
Description: Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
itgvallem.1  |-  ( _i
^ K )  =  T
Assertion
Ref Expression
itgvallem  |-  ( k  =  K  ->  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  ( B  /  (
_i ^ k ) ) ) ) ,  ( Re `  ( B  /  ( _i ^
k ) ) ) ,  0 ) ) )  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( B  /  T ) ) ) ,  ( Re
`  ( B  /  T ) ) ,  0 ) ) ) )
Distinct variable groups:    x, k    x, K
Allowed substitution hints:    A( x, k)    B( x, k)    T( x, k)    K( k)

Proof of Theorem itgvallem
StepHypRef Expression
1 oveq2 5866 . . . . . . . . 9  |-  ( k  =  K  ->  (
_i ^ k )  =  ( _i ^ K ) )
2 itgvallem.1 . . . . . . . . 9  |-  ( _i
^ K )  =  T
31, 2syl6eq 2331 . . . . . . . 8  |-  ( k  =  K  ->  (
_i ^ k )  =  T )
43oveq2d 5874 . . . . . . 7  |-  ( k  =  K  ->  ( B  /  ( _i ^
k ) )  =  ( B  /  T
) )
54fveq2d 5529 . . . . . 6  |-  ( k  =  K  ->  (
Re `  ( B  /  ( _i ^
k ) ) )  =  ( Re `  ( B  /  T
) ) )
65breq2d 4035 . . . . 5  |-  ( k  =  K  ->  (
0  <_  ( Re `  ( B  /  (
_i ^ k ) ) )  <->  0  <_  ( Re `  ( B  /  T ) ) ) )
76anbi2d 684 . . . 4  |-  ( k  =  K  ->  (
( x  e.  A  /\  0  <_  ( Re
`  ( B  / 
( _i ^ k
) ) ) )  <-> 
( x  e.  A  /\  0  <_  ( Re
`  ( B  /  T ) ) ) ) )
8 eqidd 2284 . . . 4  |-  ( k  =  K  ->  0  =  0 )
97, 5, 8ifbieq12d 3587 . . 3  |-  ( k  =  K  ->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( B  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( B  / 
( _i ^ k
) ) ) ,  0 )  =  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( B  /  T ) ) ) ,  ( Re
`  ( B  /  T ) ) ,  0 ) )
109mpteq2dv 4107 . 2  |-  ( k  =  K  ->  (
x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( B  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( B  / 
( _i ^ k
) ) ) ,  0 ) )  =  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( B  /  T ) ) ) ,  ( Re
`  ( B  /  T ) ) ,  0 ) ) )
1110fveq2d 5529 1  |-  ( k  =  K  ->  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  ( B  /  (
_i ^ k ) ) ) ) ,  ( Re `  ( B  /  ( _i ^
k ) ) ) ,  0 ) ) )  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( B  /  T ) ) ) ,  ( Re
`  ( B  /  T ) ) ,  0 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   _ici 8739    <_ cle 8868    / cdiv 9423   ^cexp 11104   Recre 11582   S.2citg2 18971
This theorem is referenced by:  iblcnlem1  19142  itgcnlem  19144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-ov 5861
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