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Theorem mbfi1fseqlem2 19071
Description: Lemma for mbfi1fseq 19076. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1  |-  ( ph  ->  F  e. MblFn )
mbfi1fseq.2  |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )
mbfi1fseq.3  |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) ) )
mbfi1fseq.4  |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m J x )  <_  m , 
( m J x ) ,  m ) ,  0 ) ) )
Assertion
Ref Expression
mbfi1fseqlem2  |-  ( A  e.  NN  ->  ( G `  A )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A
) ,  0 ) ) )
Distinct variable groups:    x, m, y, F    x, G    m, J    ph, m, x, y    A, m, x, y
Allowed substitution hints:    G( y, m)    J( x, y)

Proof of Theorem mbfi1fseqlem2
StepHypRef Expression
1 negeq 9044 . . . . . 6  |-  ( m  =  A  ->  -u m  =  -u A )
2 id 19 . . . . . 6  |-  ( m  =  A  ->  m  =  A )
31, 2oveq12d 5876 . . . . 5  |-  ( m  =  A  ->  ( -u m [,] m )  =  ( -u A [,] A ) )
43eleq2d 2350 . . . 4  |-  ( m  =  A  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u A [,] A
) ) )
5 oveq1 5865 . . . . . 6  |-  ( m  =  A  ->  (
m J x )  =  ( A J x ) )
65, 2breq12d 4036 . . . . 5  |-  ( m  =  A  ->  (
( m J x )  <_  m  <->  ( A J x )  <_  A ) )
76, 5, 2ifbieq12d 3587 . . . 4  |-  ( m  =  A  ->  if ( ( m J x )  <_  m ,  ( m J x ) ,  m
)  =  if ( ( A J x )  <_  A , 
( A J x ) ,  A ) )
8 eqidd 2284 . . . 4  |-  ( m  =  A  ->  0  =  0 )
94, 7, 8ifbieq12d 3587 . . 3  |-  ( m  =  A  ->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x )  <_  m ,  ( m J x ) ,  m ) ,  0 )  =  if ( x  e.  (
-u A [,] A
) ,  if ( ( A J x )  <_  A , 
( A J x ) ,  A ) ,  0 ) )
109mpteq2dv 4107 . 2  |-  ( m  =  A  ->  (
x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m J x )  <_  m , 
( m J x ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A
) ,  0 ) ) )
11 mbfi1fseq.4 . 2  |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m J x )  <_  m , 
( m J x ) ,  m ) ,  0 ) ) )
12 reex 8828 . . 3  |-  RR  e.  _V
1312mptex 5746 . 2  |-  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A ) ,  0 ) )  e.  _V
1410, 11, 13fvmpt 5602 1  |-  ( A  e.  NN  ->  ( G `  A )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A
) ,  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   0cc0 8737    x. cmul 8742    +oocpnf 8864    <_ cle 8868   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   [,)cico 10658   [,]cicc 10659   |_cfl 10924   ^cexp 11104  MblFncmbf 18969
This theorem is referenced by:  mbfi1fseqlem3  19072  mbfi1fseqlem4  19073  mbfi1fseqlem5  19074  mbfi1fseqlem6  19075
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-neg 9040
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