MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itgvallem Structured version   Unicode version

Theorem itgvallem 19676
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 6089 . . . . . . . . 9  |-  ( k  =  K  ->  (
_i ^ k )  =  ( _i ^ K ) )
2 itgvallem.1 . . . . . . . . 9  |-  ( _i
^ K )  =  T
31, 2syl6eq 2484 . . . . . . . 8  |-  ( k  =  K  ->  (
_i ^ k )  =  T )
43oveq2d 6097 . . . . . . 7  |-  ( k  =  K  ->  ( B  /  ( _i ^
k ) )  =  ( B  /  T
) )
54fveq2d 5732 . . . . . 6  |-  ( k  =  K  ->  (
Re `  ( B  /  ( _i ^
k ) ) )  =  ( Re `  ( B  /  T
) ) )
65breq2d 4224 . . . . 5  |-  ( k  =  K  ->  (
0  <_  ( Re `  ( B  /  (
_i ^ k ) ) )  <->  0  <_  ( Re `  ( B  /  T ) ) ) )
76anbi2d 685 . . . 4  |-  ( k  =  K  ->  (
( x  e.  A  /\  0  <_  ( Re
`  ( B  / 
( _i ^ k
) ) ) )  <-> 
( x  e.  A  /\  0  <_  ( Re
`  ( B  /  T ) ) ) ) )
8 eqidd 2437 . . . 4  |-  ( k  =  K  ->  0  =  0 )
97, 5, 8ifbieq12d 3761 . . 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 4296 . 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 5732 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 359    = wceq 1652    e. wcel 1725   ifcif 3739   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   _ici 8992    <_ cle 9121    / cdiv 9677   ^cexp 11382   Recre 11902   S.2citg2 19508
This theorem is referenced by:  iblcnlem1  19679  itgcnlem  19681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-iota 5418  df-fv 5462  df-ov 6084
  Copyright terms: Public domain W3C validator