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Theorem itgvallem 19155
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 5882 . . . . . . . . 9  |-  ( k  =  K  ->  (
_i ^ k )  =  ( _i ^ K ) )
2 itgvallem.1 . . . . . . . . 9  |-  ( _i
^ K )  =  T
31, 2syl6eq 2344 . . . . . . . 8  |-  ( k  =  K  ->  (
_i ^ k )  =  T )
43oveq2d 5890 . . . . . . 7  |-  ( k  =  K  ->  ( B  /  ( _i ^
k ) )  =  ( B  /  T
) )
54fveq2d 5545 . . . . . 6  |-  ( k  =  K  ->  (
Re `  ( B  /  ( _i ^
k ) ) )  =  ( Re `  ( B  /  T
) ) )
65breq2d 4051 . . . . 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 2297 . . . 4  |-  ( k  =  K  ->  0  =  0 )
97, 5, 8ifbieq12d 3600 . . 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 4123 . 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 5545 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 1632    e. wcel 1696   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   _ici 8755    <_ cle 8884    / cdiv 9439   ^cexp 11120   Recre 11598   S.2citg2 18987
This theorem is referenced by:  iblcnlem1  19158  itgcnlem  19160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-iota 5235  df-fv 5279  df-ov 5877
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